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GIFT  or 


SOLID   GEOMETRY 


A  SERIES  OF  MATHEMATICAL  TEXTS 

EDITED    BY 

EARLE  RAYMOND  HEDRICK 


THE  CALCULUS 

By    Ellery    Williams    Davis    and    William    Charles 
Brenke. 

PLANE   AND    SOLID   ANALYTIC   GEOMETRY 

By  Alexander  Ziwet  and  Louis  Allen  Hopkins. 

PLANE     AND     SPHERICAL     TRIGONOMETRY     WITH 
COMPLETE   TABLES 
By  Alfred  Monroe  Kenyon  and  Louis  Ingold. 

PLANE     AND     SPHERICAL     TRIGONOMETRY     WITH 
BRIEF  TABLES 
By  Alfred  Monroe  Kenyon  and  Louis  Ingold. 

THE   MACMILLAN  TABLES 

Prepared  under  the  direction  of  Earle  Raymond  Hedrick. 

PLANE   GEOMETRY 

By  Walter  Burton  Ford  and  Charles  Ammerman. 

PLANE   AND   SOLID   GEOMETRY 

By  Walter  Burton  Ford  and  Charles  Ammerman. 

SOLID   GEOMETRY 

By  Walter  Burton  Ford  and  Charles  Ammerman. 


SOLID    GEOMETRY 


BY 

WALTER  BURTON   FORD 

JUNIOR   PROFESSOR   OF   MATHEMATICS,    THE    UNIVERSITY    OF    MICHIGAN 

AND 

CHARLES   AMMERMAN 

THE   WILLIAM    MCKINLEY    HIGH    SCHOOL,    ST.    LOUIS 


EDITED    BY 

EARLE  RAYMOND  HEDRICfC 


!c 


THE   MACMILLAN   COMPANY 
1920 

All  rights  reserved 


Copyright,  1913, 
By  the   MACMILLAN  COMPANY. 


Set  up  and  electrotyped.     Published  November,  1913. 


J.  8.  Cushing  Co.  —  Berwick  &  Smith  Co. 
Norwood,  Mass.,  U.S.A. 


PREFACE 

This  book  contains  the  Chapters  on  Solid  Geometry  from 
the  Plane  and  Solid  Geometry  by  the  same  authors.  The 
general  nature  of  the  motives  that  led  to  the  organization  of 
the  work  are  described  in  the  preface  of  the  complete  edition, 
and  it  does  not  seem  necessary  to  repeat  all  of  them  here. 

In  order  to  make  it  possible  to  refer  to  theorems  proved  in 
Plane  Geometry,  a  complete  syllabus  of  them,  together  with 
other  necessary  quotations,  is  printed  on  pages  xxix-xlvi  of  this 
book.  All  references  made  in  the  text,  and  any  other  questions 
in  Plane  Geometry  concerning  which  there  may  be  doubt,  can 
there  be  looked  up  by  the  student.  An  excellent  opportunity 
for  a  review  of  Plane  Geometry  is  afforded  by  this  syllabus. 

The  book  is  distinguished  by  its  acceptance  of  the  principle 
of  emphasis  of  important  theorems  laid  down  by  the  Commit- 
tee of  Fifteen  of  the  National  Education  Association  in  their 
Keport.*  Thus,  theorems  of  the  greatest  value  and  importance 
are  printed  in  bold-faced  type,  and  those  whose  importance  is 
considerable  are  printed  in  large  italics. 

The  Report  just  mentioned  has  been  of  great  assistance,  and 
its  principles  have  been  accepted  in  general,  not  in  a  slavish 
sense  but  in  the  broad  manner  recommended  by  the  Committee 
itself.  A  perusal  of  the  Eeport  will  give  more  fully  and 
accurately  than  could  be  done  in  this  brief  preface,  the  con- 
siderations which  led  to  the  adoption  of  these  principles,  in 
particular,  the  principle  of  emphasis  upon  important  theorems, 
both  by  the  Committee  and  by  the  authors  of  this  book. 


*  Printed  as  a  separate  pamphlet  with  the  Proceedings  for  1912.  Ke- 
printed  also  in  School  Science,  1911,  and  in  The  Mathematics  Teacher, 
December,  1912. 

V 


459948 


vi  PREFACE 

The  great  excellence  of  the  figures,  particularly  the  very 
unusual  and  effective  *  phantom '  halftone  engravings,  deserves 
mention.  These  figures  should  go  far  toward  relieving  the 
unreality  which  often  attaches  to  the  constructions  of  Solid 
Geometry  in  the  minds  of  students. 

W.  B.  FORD, 
CHAS.  AMMERMAN, 
E.  R.  HEDRICK,  Editor. 


CONTENTS 

Chapter  VI.    Lines  and  Planes  in  Space   . 
Part  1.       General  Principles 
Part  II.     Perpendiculars  and  Parallels 
Part  III.   Dihedral  Angles    . 
Part  IV.    Polyhedral  Angles 
Miscellaneous  Exercises  on  Chapter  VI 


Chapter  VII.    Polyhedrons.    Cylinders.    Cones 
Part  I.       Prisms.     Parallelepipeds 

Part  II.     Pyramids 

Part  III.   Cylinders  and  Cones      .... 
Part  IV.    General  Theorems  on  Polyhedrons.   Similar- 
ity.    Regular  Solids.     Volumes 
Miscellaneous  Exercises  on  Chapter  VII 


Chapter  VIII.    The  Sphere 

Part  I.       General  Properties         .... 
Part  II.     Spherical  Angles.     Triangles.     Polygons 
Part  III.    Areas  and  Volumes       .... 
Miscellaneous  Exercises  on  Chapter  VIII    . 


Tables 

Table  I.       Quantities  Determined  by  a  Given  Angle 
Table  II.     Powers  and  Roots        .... 
Table  III.   Important  Numbers    .... 


Syllabus  of  Plane  Geometry 

Introduction  .... 
Chapter  I.       Rectilinear  Figures 
Chapter  II.     The  Circle 


PAGES 

215-237 

215-218 
219-227 
228-231 
232-234 
235-237 

238-283 

238-252 
253-262 
263-273 

274-281 

282-283 

284-321 

284-293 
296-308 
309-317 
317-321 

i-xxvii 

ii-vii 

viii-xxvi 

xxvii 

xxix-xlvi 

XX  ix 

xxxii 

xxxvii 


CONTENTS 


IX 


Chapter  III.   Proportion.     Similarity    . 

Chapter  IV.    Areas  of  Polygons.     Pythagorean  Theo 

rem 

Chapter  V.      Regular  Polygons  and  Circles  . 
Appendix  :      Maxima  and  Minima 


Index 


xliii 
xliv 
xlv 

xlvii-xlix 


The  complete  text  for  the  theorems 
listed  in  the  Syllabus  of  Theorems 
of  Plane  Geometry  in  this  book  is 
published  separately  under  the  title 

PLANE  GEOMETRY 

The  contents  of  that  book  and  of  this 
one  are  published  together  under  the 
title 

PLANE  AND  SOLID  GEOMETRY 


!« 


SOLID   GEOMETRY 

CHAPTER   Yl 

LINES   AND   PLANES   IN   SPACE 

PAET   I.     GENERAL   PRINCIPLES 

239.  Definitions.  Solid  Geometry,  or  Geometry  of  Three  Di- 
mensions, treats  of  figures  whose  parts  are  not  confined  to  a 
plane. 

A  plane  is  a  surface  such  that  if  any  two  points  in  it  are 
taken,  the  straight  line  passing  through  them  lies  wholly  in 
the  surface. 


Fig.  163 

Thus,  m  Fig.  163,  if  A  and  B  are  two  points  of  a  plane  iHfiV,  the 
entire  straight  line  AB  lies  in  the  plane  MN.  Any  point  O  on  AB  lies 
inMN. 

A  plane  is  said  to  be  determined  by  certain  points  and  lines 
if  that  plane  and  no  other  plane  contains  those  points  and 
lines. 

240.  Corollary  1.  It  is  evident  from  the  definition  of  a 
plane  that  if  a  line  has  two  of  its  points  in  a  plane^  it  lies  wholly 
in  that  plane. 

216 


216 


LINES  AND   PLANES  IN  SPACE       [VI,  §  241 


241.  Assumptions,  or  Postulates. 

1.  A  plane  is  unlimited  in  extent. 

2.  TJirough  any  straight  line  an 
unliinited  number  of  planes  may 
he  passed.     See  Fig.  164. 

3.  If  a  plane  is  revolved  about 
any  straight  Ime  in  it  as  an  axis, 
it  may  be  made  to  pass  through 
any  point  in  space. 

4.  One  and  only  one  plane  can  he  made  to  pass  through  three 
points  not  in  the  same  straight  line. 


Fig.  164 


Fig.  165  (a) 


Fig.  165  (&) 


Fig.  165  {a)  represents  a  plane  PQ  through  three  points  A,  B,  C. 
Fig.  165  (6)  represents  a  plane  piece  of  glass  resting  on  the  points  of 
three  tacks. 

5.    Two  planes  cannot  intersect  each  other  in  only  a  single  point. 

242.  Corollary  1.  A  plane  is  determined  by  tiuo  intersecting 
lines. 


Fig.  166  (a) 


Fig.  166  (&) 


[Hint.     Consider  the  point  where  the  lines  intersect,  and  two  other 
points,  one  on  each  line  ;  then  apply  4,  §  241.] 


VI,  §  244] 


GENERAL  PRINCIPLES 


217 


243.    Corollary  2.     A  line  and  a  point  ivithout  the  line  de- 
termine a  plane. 

[Hint.     Use  4,  §  241.]  [^^ '  ! 


Fig.  167  (a)  Fia.  167  (&) 

244.   Corollary  3-     Two  parallel  lines  determine  a  plane. 


Fig.  168  (a) 


Fig.  168  (6) 


[Hint.     By  the  definition  of  parallel  lines  (§  48),  two  such  lines  must 
lie  in  a  plane.     Show  that  this  is  the  only  one.] 


EXERCISES 

1.  How  many  planes  pass  through  a  given  straight  line  in 
space  ?     How  many  pass  through  two  given  points  ? 

2.  In  a  carpenter's  plane  the  knife-edge  lies  along  a  straight 
line.  As  soon  as  any  rough  surface  has  been  sufficiently  planed 
off,  the  whole  length  of  the  knife-edge  keeps  on  the  surface  as 
the  plane  is  moved  along.     Connect  this  fact  with  §  240. 

3.  Why  are  cameras,  surveyors'  transits,  etc.,  mounted  on 
three  legs  instead  of  four  ? 

4.  Prove  that  a  straight  line  can  intersect  a  plane  in  but  one 
point  unless  it  lies  wholly  in  the  plane.     See  §  240. 


!i 

0^' 

Fig.  169 


218  LINES  AND  PLANES  IN  SPACE       [VI,  §  245 

245.  Theorem  I.  The  intersection  of  two  ^ilanes  is  a  straight 
line. 

Given  the  two  intersecting  planes  MN 
and  RS. 

To  prove  that  their   intersection   is   a 
straight  line. 

Proof.     Let  A  and  B  be  any  two  points 
common  to  both  planes.  5,  §  241 

Draw  the  straight  line  AB. 

Then  every  point  in  AB  lies  in  MN  and  also  in  RS.  §  240 

Therefore,  AB  is  common  to  the  two  planes. 

Moreover,  no  point  not  on  AB  can  be  common  to  both  planes, 
for  the  two  planes  would  then  coincide.  4,  §  241 

Therefore,  the  intersection  of  the  planes  MN  and  RS  is  a 
straight  line. 

EXERCISES 

1.  What  is  the  locus  of  all  points  common  to  two  intersect- 
ing planes  ? 

2.  If  a  sheet  of  paper  is  folded,  why  is  the  crease  straight  ? 

3.  In  how  many  points  (in  general)  will  three  planes  inter- 
sect? What  can  be  said  of  the  intersection  of  four  or  more 
planes  in  space  ? 

4.  Can  two  pencils  be  held  in  such  a  position  that  a  plane 
cannot  be  passed  through  them  ?  State  the  general  fact  about 
a  plane  that  is  illustrated  by  your  answer. 

5.  Can  a  plane  be  passed  (in  general)  through  four  or  more 
given  points  in  space?  Can  a  plane  be  passed  (in  general) 
through  three  lines  all  of  which  pass  through  a  common  point 
in  space  ? 

6.  Can  there  be  two  straight  lines  that  are  not  parallel  and 
that  do  not  meet  ?     Find  a  pair  of  such  lines  in  Fig.  169. 


VI,  §248]    PERPENDICULARS  AND   PARALLELS  219 

PART   11.     PERPENDICULARS   AND   PARALLELS 

246.  Line  Perpendicular  to  a  Plane.  The  point  where  a 
line  intersects  a  plane  is  called  the  foot  of  the  line  on  that 
plane. 

A  straight  line  is  perpendicular  to  a  plane  when  it  is  perpen- 
dicular to  every  straight  line  in  the  plane  drawn  through  its 


Fig.  170  (a) 


Fig.  170  (6) 


foot.  The  plane  is  then  also  said  to  be  perpendicular  to  the 
line.  Thus,  in  Fig.  170  (a),  if  PQ  is  perpendicular  to  the  plane 
MJ^j  it  is  then  perpendicular  to  all  the  lines  QA,  QB,  QC,  etc. ; 
and  PQ  is  called  the  distance  from  P  to  MN]  see  Ex.  1  below. 

247.  Parallel  Planes  and  Lines.  A  straight  line  is  parallel 
to  a  plane  if  they  never  meet,  however  far  produced.  Two 
planes  are  parallel  if  they  never  meet,  however  far  produced. 

It  is  to  be  remembered  (§  48)  that  two  lines  are  parallel  only 
when  they  lie  in  the  same  plane  and  do  not  meet. 

248.  Corollary  1.  A  plane  that  contains  one  of  two  parallel 
lines  is  parallel  to  the  other  line. 


EXERCISES 

1.  Show,  by  §  77,  that  the  perpendicular  from  a  point  P  to 
a  plane  MN  (Fig.  170  a)  is  shorter  than  any  other  line  that 
can  be  drawn  from  P  to  MN. 

2.  Show,  by  §  71,  that  if  two  oblique  lines  from  a  point  P 
to  a  plane  MN  cut  off  equal  distances  from  the  foot  of  the  per- 
pendicular from  P  to  MN,  they  are  equal.     See  Ex.  1,  p.  63. 


220 


LINES  AND   PLANES   IN   SPACE        [VI,  §  249 


Fig.  171 


249.  Theorem  II.  If  a  line  is  perpendicular  to  each  of  two 
lines  at  their  point  of  intersection,  it  is  perpendicular  to  their 
plane. 

Given  FB  perpendicular  at  B 
to  each  of  two  straight  lines  AB 
and  BC  ot  the  plane  MN. 

To  prove  FB  perpendicular  to 
the  plane  MJ^. 

Proof.  Draw  A  C,  and  through 
B  draw  any  line,  as  BH,  meet- 
ing AC  Sit  II. 

Prolong  FB  to  E  so  that 
BE  =  FB.     Join  F  and  E  to  A,  H,  and  O. 

Then  AB  and  BG  are  perpendicular  bisectors  of  FE 
whence  FA  =  AE,  FG  =  GE. 

Therefore  A  AFG  ^  A  AEG, 

whence  Z  HAF  =  Z  HAE. 

Also  A  HAF  ^  A  HAE, 

and  HF=HE. 

Hence  HB±FE  ov  FB. 

But  ^5  was  any  line  in  MN'  drawn  through  ^. 

Therefore  FBA.MK 


Const. 

§100 

§45 

Why? 

Why? 

Why? 

AVhy? 


§246 


250.    Corollary  1.     At  a  point  in  a  plane  only  one  perpen- 
dicular line  can  he  erected. 

[Hint,  Suppose  a  second  perpen- 
dicular line  BC  could  be  erected  (Fig. 
172).  Pass  a  plane  through  AB  and 
BC.  This  plane  will  intersect  MN'm.  a 
straight  line,  as  DE.  Then  AB  and 
BC  are  both  perpendicular  to  DE  at  the 
same  point  B.  But,  since  AB,  BC,  and  DE  ail  lie  in  the  same  plane, 
this  is  impossible,  by  7,  §  31.] 


Fig.  172 


VI,  §254]    PERPENDICULARS  AND   PARALLELS 


221 


251.   Corollary  2.     From  a  point  icithout  a  plane,  only  one 
line  can  be  drawn  perpendicular  to  the 
plane. 

[Hint.  If  two  perpendiculars,  as  PB 
and  PA,  could  be  drawn  from  P  to  the 
plane  MN,  then  A  PBA  would  contain 
two  right  angles  so  that  the  sum  of  the 
angles  of  APBA  would  be  more  than 
two  right  angles.  But  this  is  impossible. 
Why  ?] 


Fig.  173 


252.  Corollary  3.  Through  a  given  point  in  a  straight  line, 
only  one  plane  can  he  drawn  perpendicular  to  the  line. 

[Hint.  Draw  two  different  perpendiculars  in  space  to  the  given  line 
at  the  given  point,  and  apply  §§  242,  249.  If  two  such  planes  exist,  their 
intersections  with  a  plane  through  the  given  line  violate  7,  §  31.] 

253.  Corollary  4.  TJirough  a  given  point  without  a  straight 
line,  only  one  plane  can  he  drawn  perpendicular  to  the  line. 

[Hint.  Prove  by  reduction  to  an  absurdity.  Show  that  the  inter- 
sections of  two  sucb  perpendicular  planes  with  the  plane  determined  by 
the  given  line  and  given  point  would  violate  §  58.] 

254.  Corollary  5.  All  perpendicular  lines  that  can  he  drawn 
to  a  straight  line  at  a  given  point  in  it  lie  in  a  plane  perpendicular 
to  the  line  at  the  given  point. 

[Hint.  Show  that  otherwise  two  perpendicular  lines  could  be  drawn 
to  the  given  line  in  the  same  plane  at  the  given  point,  thus  violat- 
ing 7,  §31.] 

EXERCISES 

1.  Show  how  to  determine  a  perpendicular  to  a  plane  by 
means  of  two  carpenter's  squares. 

2.  Tell  how  to  test  whether  or  not  a  flagpole  is  erect. 

3.  A  spoke  of  a  wheel  is  perpendicular  to  the  axis  on  which 
it  turns.  Show  by  §  254  that  it  describes  a  plane  in  its 
rotation. 


222 


LINES  AND  PLANES  IN  SPACE       [VI,  §  255 


255.   Theorem  III.     Two  planes  per- 
pendicular to  the  same  line  are  parallel. 

[Hint.     Show  that  if  the  two  planes  met, 
say  in  a  point  P,  §  253  would  be  violated.] 


256.  Theorem  IV.  If  a  plane 
intersects  two  parallel  planes,  the 
lilies  of  intersection  are  parallel. 

Given  the  plane  PQ  intersect- 
ing the  parallel  planes  MN  and 
ES  in  AB  and  CD,  respectively. 

To  prove    AB  II  CD. 

[Hint.  Prove,  by  reduction  to  an 
absurdity,  that  AB  and  CD  cannot 
meet.] 


M 

f 

R< 

1 

?u 

7' 

U< 

^ 

^ 

.../"■^r-^ 

r 

Fig.  175 


257.  Theorem  V.  •  Two  lines  parallel  to  a  third  line  (in 
space)  are  parallel  to  each  other.     Compare  §  50. 

[Hint.  Let  BB'  and  CC  be  two  lines  parallel  to  a  third  line  AA' 
(Fig.  176).  The  plane  determined  by  OC  and  the  point  B  on  BB'  is  par- 
allel to  AA'  (§  248) .  Therefore  (§  48)  the  line  of  intersection  of  this  plane 
with  the  plane  of  the  parallels  AA'  and  BB'  is  parallel  to  AA'.  Hence 
show,  by  §  49,  that  this  line  of  intersection  coincides  with  BB'^  so  that 
BB'  and  CC  lie  in  a  plane.  Finally,  show  that  BB'  and  CC  cannot 
meet ;  for,  if  they  did  meet,  say  at  a  point  D,  the  plane  determined  by 
I)  and  AA'  would  contain  (§  244)  both  BB'  and  CC.^ 


VI,  §258]    PERPENDICULARS  AND  PARALLELS 


223 


258.  Theorem  VI.  If  two  angles,  not  in  the  same  plarie,  have 
their  sides  respectively  imraUel  and  extending  in  the  same  direc- 
tion, they  are  equal  and  their  planes  are  parallel. 


Fig.  176 

Given  the  angles  BAG  and  B'AC,  lying  in  the  planes  MN 
and  PQ,  respectively,  with  AB  II  A'B',  and  AC  II  A'C. 

To  prove  that   Z.A  =  AAl,  and  that  JfA^II  PQ. 

Proof.     Take  AB  =  A'B',  and  AG  =  A'G'. 

Draw  AA',  BB',  GG\  GB,  and  C'jS'. 

Since  AB  is  equal  and  parallel  to  A'B', 
it  follows  that  ABB' A  is  a  parallelogram  ; 
hence  AA  is  equal  and  parallel  to  BB'. 

Similarly,  A  A'  is  equal  and  parallel  to  CO'. 

Hence         BB^  is  equal  and  parallel  to  GG'. 

Then  BB'C'G  is  a  parallelogram,  and  GB=G'B'. 

Therefore  A  ABG  ^  A  A'B'G'. 

Hence  Z.A  =  ZA'. 

Now  PQ  II  AB.     Likewise  PQ  II  AG. 

Therefore,  PQ  II  ilOT"  for,  if  not,  the  line  of  intersection  of 
PQ  and  MN  would  meet  either  AB  ot  AG  (or  both)  extended ; 
hence  PQ  would  not  be  parallel  to  each  of  them. 

Note.  The  similar  theorem  for  angles  that  lie  in  the  same  plane  was 
proved  in  §  67.  As  in  §  67,  the  two  angles  are  supplementary  to  each 
other  if  one  pair  of  corresponding  sides  extend  in  opposite  directions  from 
the  vertices. 


Why? 

Why? 

§257 
Why? 
Why? 
Why? 

§248 


224 


LINES  AND  PLANES  IN  SPACE       [VI,  §  259 


Fig.  177 


259.    Theorem  VII.     A  plane  perpendicular  to   one 
parallel  lines  is  perpendicular  to  the  other  also. 

Given  the  two  parallel  lines  AB 
and  CD,  and  a  plane  MN  perpen- 
dicular to  CD  at  C. 

To  prove  that  MN  is  perpendicu- 
lar to  AB. 

Proof.  The  parallel  lines  AB 
and  CD  determine  a  plane  (§  244) 
which  intersects  MN  in  some  line  AC. 

'Now AC  is  perpendicular  to  CD  ; 
whence  ACis  perpendicular  to  AB. 

Draw  any  line  AE  in  the  plane  Jf A'' through  A. 

Draw  CF  in  MlSf  parallel  to  AE  through  C. 

Then  CF  is  perpendicular  to  CD. 

Hence  AE  is  perpendicular  to  AB. 

Therefore  AB  is  perpendicular  to  MN". 


of  two 


§245 

§246 

§60 


§  246 

§  258 
§246 


260.   Corollary  1.     Two  lines  perpendicular  to  the  same  plane, 
are  parallel. 

[Hint.  Let  AB  and  CD  (Fig.  177)  be  perpendicular  to  the  plane  Mli. 
Imagine  a  parallel  CD'  to  AB  through  C.  Then  CD'  is  perpendicular  to 
MN,  by  §  259.     Hence  CD'  coincides  with  CD,  by  §  250.] 


EXERCISES 

1.  The  legs  of  a  table  lie  along  parallel  lines  in  space.  What 
preceding  theorem  or  corollary  is  illustrated  here  ?  Mention 
other  similar  illustrations. 

2.  How  many  lines  can  be  drawn  through  a  given  point 
parallel  to  a  given  plane  ?  If  there  is  more  than  one  such,  what 
is  the  locus  of  them  all  ?  • 

3.  Given  a  plane  and  two  points  without  it.  When  will  the 
line  through  the  two  points  be  parallel  to  the  plane  ? 


VI,  §261]    PERPENDICULARS  AND  PARALLELS 


225 


261.  Theorem  VIII.  If  tioo  straight  lines  are  inter- 
sected by  three  parallel  planes,  the  corresponding  seg- 
ments of  these  lines  are  proportional. 

Given  the  straight  lines 
AB  and  CD  cut  by  the 


parallel 

and  N, 


planes     L,    M, 


To  prove  that 
AE/EB  =  GF/FD. 

Proof.  Draw  BQ  meet- 
ing the  plane  M  in  G. 
Draw  EF,  EG,  FG,  BD, 
and  AC. 

Then    GF  II   BD,   and  EG  11  AC 

Now      AE/EB=CG/GB,     and     CF/FD 

Therefore  AE/EB  =  CF/FD. 

EXERCISES 


Fig.  178 


§  256 

CG/GB.   §145 

Why? 


1.  Show  that  if  parallel  planes  intercept  equal  segments  on 
one  line,  they  will  intercept  equal  segments  on  any  other  line. 

2.  In  Fig.   178,   AE  =  5,   EB  =  4:,   and    CF=6.     What  is 
the  value  of  FD  ? 

3.  Two  ordinary  blocks  C 
and  D  having  the  respective 
heights  H  and  h  are  placed 
upon  each  other  as  shown  in 
the  figure.  Show  that  any  line 
AB  drawn  from  the  upper  sur- 
face of  C  to  the  lower  surface 
of  D  will  be  divided  in  the 
ratio  H'.hhj  the  point  P  where 
AB  intersects  the  common  surface  of  the  two  blocks. 

Q 


1 

/4,  ^, 

He 

■'■  .\ 

—/ 

--     — f 

V 

226 


LINES  AND  PLANES  IN  SPACE       [VI,  §  262 


262.  Perpendicular  Planes.  Two  planes  MN  and  PQ  are 
said  to  be  perpendicular  to  each  other  when  any  line  CD  drawn 
in  the  one  perpendicular  to  their  intersection  is  perpendicular 
to  the  other  plane. 


^ 

M<C,^ 

1 

}           r 

O 

Fig. 

179. 

263.  Theorem  IX.  If  a  straight  line  is  perpendicular 
to  a  plane,  every  plane  containing  this  line  is  perpen- 
dicular to  the  given  plane. 

Given  the  line  CD  perpendicular  to  plane  MN]  and  given 
any  plane  PQ  containing  the  line  CD. 

To  prove  that  plane  PQ  is  perpendicular  to  plane  MN. 

Proof.  Let  AB  be  the  line  of  intersection  of  the  two  planes 
JO/"  and  PQ.  Imagine  any  line  CD'  in  the  plane  PQ  perpen- 
dicular to  the  line  AB. 

Then  CD  is  parallel  to  OD".  §  52 

But  CD  is  perpendicular  to  MJSf-,  Given 

hence  CD'  is  perpendicular  to  MN.  §  259 

Since  CD'  is  any  line  of  the  plane  PQ  perpendicular  to  AB, 
it  follows  that  PQ  is  perpendicular  to  MN.  §  262 

264.  Corollary  1.  The  line  perpendicular  to  a  given  plane 
at  a  given  point  lies  in  any  plane  through  that  point  perpendicular 
to  the  given  plane. 


VI,  §  265]    PERPENDICULARS  AND   PARALLELS  227 

265.  Theorem  X.  If  each  of  two  intersecting  planes  is  per- 
pendicular to  a  third  plane,  their  line  of  intersection  is  perpen- 
dicular to  the  third  plane. 


Fig.  180 

Given  the  planes  PQ  and  RS  perpendicular  to  plane  MN 
and  intersecting  each  other  in  AB. 

To  prove  that  AB  is  perpendicular  to  JOT". 

Proof.  Suppose  that  AB  is  not  perpendicular  to  JOT,  but 
that  some  other  line  as  OF  through  C,  the  point  common  to  the 
three  planes,  is  the  perpendicular  to  MN. 

Then  OF  lies  in  RS  and  in  PQ.  §  264 

Hence  OF  coincides  with  AB.  §  245 

Therefore  AB  is  perpendicular  to  MN  at  C. 


EXERCISES 

1.  The  blades  of  a  side  paddle  wheel  of  a  steamboat  are  all 
perpendicular  to  the  side  of  the  boat.  Connect  this  fact  with 
one  of  the  preceding  theorems.  Do  the  same  with  the  fact 
that  the  upright  edge  of  any  building  is  vertical. 

2.  How  many  planes  can  be  drawn  perpendicular  to  a  given 
plane  and  passing  through  a  given  line  in  space? 

[Hint.  Select  a  point  in  the  given  line,  draw  the  perpendicular  line 
through  that  point  to  the  given  plane,  and  consider  all  the  planes  that  can 
be  passed  through  this  perpendicular.] 


228  LINES  AND   PLANES   IN  SPACE       [VI,  §  266 

PAET   IIL     DIHEDRAL   ANGLES 

266.  Dihedral  Angles.  The  figure  formed  by  two  inter- 
secting portions  of  planes  bounded  by  their  line  of  intersection 
is  called  a  dihedral  angle.  The  planes  forming  the  dihedral 
angle  are  its  faces  and  the  line  of  intersection  is  its  edge. 

A  dihedral  angle  may  be  designated  by  the  two  letters  on 
its  edge,  or  by  the  two  letters  on  its  edge  together  with  an 
additional  letter  on  each  face. 

C 


Fig.  181 


Thus,  in  the  figure,  the  planes  ^Dand  AS  are  the  faces  and 
AG  is  the  edge  of  the  dihedral  angle  B-CA-D. 

The  plane  angle  of  a  dihedral  angle  is  an  angle  formed  by 
lines  in  the  two  faces  perpendicular  to  the  edge  at  the  same 
point.  Thus,  GFE  is  the  plane  angle  of  the  dihedral  angle 
B-CA-D. 

The  magnitude  of  a  dihedral  angle  does  not  depend  upon 
the  extent  of  its  faces.  If  a  plane  be  made  to  revolve  from 
the  position  of  one  face  about  the  edge  as  an  axis  to  the  posi- 
tion of  the  other  face,  it  turns  through  the  dihedral  angle,  and 
the  greater  the  amount  of  turning,  the  greater  the  angle. 

267.  Measure  of  Dihedral  Angles.  The  plane  angle  of  a 
dihedral  angle  is  taken  as  its  measure,  so  that  two  dihedral 
angles  are  always  in  the  same  ratio  as  the  magnitudes  of  their 
plane  angles.  In  particular,  two  dihedral  angles  are  equal 
when  their  plane  angles  are  equal. 


VI,  §  267] 


DIHEDRAL  ANGLES 


229 


Dihedral  angles  are  right,  acute,  obtuse,  etc.,  according  as 
their  plane  angles  are  right,  acute,  obtuse,  etc.  Similar  defi- 
nitions exist  for  complementary  dihedral  angles,  supplemen- 
tary dihedral  angles,  vertical  dihedral  angles,  etc.  The  faces 
of  a  right  dihedral   angle  are   perpendicular   to  each  other. 


EXERCISES 

1.  Read  the  adjacent  dihedral  angles  in  the  following  figure. 
Read  the  vertical,  the  complementary,  the  supplementary  di- 
hedral angles. 


2.  If  two  planes  intersect  each  other,  show  that  the  op- 
posite or  vertical  dihedral  angles  thus  formed  are  equal. 

[Hint.     Use  §  267.] 

3.  Show  that  the  dihedral  angle  through  which  a  door  is 
opened  is  measured  by  the  plane  angle  through  which  the 
bottom  edge  of  the  door  moves. 

4.  Make  an  instrument  for  meas- 
uring dihedral  angles  by  cutting 
and  folding  a  piece  of  heavy  paper 
or  cardboard  in  the  manner  shown 
in  the  figure. 

5.  What  is  the  number  of  degrees  in  one  of  the  dihedral 
angles  of  a  bay  window,  it  being  understood  that  the  bay  win- 
dow consists  of  three  equal  upright  plane  sections,  and  that 
their  bases  form  three  sides  of  a  regular  octagon  ? 


230 


LINES  AND   PLANES   IN   SPACE        [VI,  §268 


268.    Theorem    XI.     Every  point  in  a  plane  that  bisects  a  di- 
hedral angle  is  equidistant  from  the  faces  of  the  angle. 


""  A^ 

i 

i 

Ja 

P 

S      / 

€y 

N, 

^y 

\| 

^ 

Fig.  182 


Given  the  plane  TR  bisecting  the  dihedral  angle  formed  by 
the  planes  TM  and  TN^  so  that  the  dihedral  angles  M-TS-R 
and  N-TS-R  are  equal ;  and  given  PA  and  PB  perpendicular 
to  TM  and  TN,  respectively,  from  any  point  P  in  TR. 

To  prove  that  PA  =  PB. 

Proof.  Pass  a  plane  through  PA  und  PB  and  let  X  be  its 
point  of  intersection  with  ST',  let  AX,  BX,  and  PX  be  the 
intersections  of  plane  PAB  with  planes  TM,  TN,  and  TR. 

Plane  PAB  ±  planes  TM  and  TN.  §  263 

Then  plane  PAB  ±  ST,  §  265 

whence  ST  1.  AX,  BX,  and  PX.  Why  ? 

The  angles  AXP  and  BXP  are  the  plane  angles  of  the 
dihedral  angles  M-ST-R  and  N-ST-R.  Why  ? 

Since  the  dihedral  angles  are  given  equal,  their  plane  angles 
are  equal,  that  is,        z.  AXP  =  Z  BXP- 

whence  rt.  A  AXP  ^  rt.  A  BXP,  Why  ? 

and  therefore  PA  =  PB.  Why  ? 


VI,  §  270]  DIHEDRAL  ANGLES  231 

269.  Corollary  1.     Any  point  not  in  the  bisecting  plane  of  a 
dihedral  angle  is  unequally  distant  from  the  two  faces. 

270.  Corollary  2.     The  plane  bisecting  a  dihedral  angle  is  the 
locus  of  all  points  equally  distant  from  the  faces  of  the  angle. 

See  Note,  §  99. 

EXERCISES 

1.  To  what  theorem  in  Plane  Geometry  does  §  268  cor- 
respond ? 

2.  From  any  point  within  a  dihedral  angle  perpendiculars 
are  drawn  to  the  faces.  Show  that  the  angle  formed  by  these 
perpendiculars  is  supplementary  to  the  plane  angle  of  the 
dihedral  angle. 

3.  Prove  that  the  two  adjacent  dihedral  angles  formed  by  one 
plane  meeting  another  are  supplementary. 

[Hint.     At  some  point  on  the  edge  of  the  dihedral,   erect  a  plane 
perpendicular  to  its  edge,  and  consider  the  plane  angles  formed.] 

4.  What  is  the  locus  of  all  points  equidistant  from  two 
intersecting  planes,  each  of  indefinite  extent  ? 

5.  What  is  the  locus  of  all  points  in  space  equidistant  from 
two  given  points  ? 

6.  What  is  the  locus  of  all  points  in  space  equidistant  from 
the  circumference  of  a  circle  ? 

7.  What  is  the  locus  in  space  of  all  points  equidistant  from 
two  intersecting  lines  ? 

8.  What  is  the  locus  of  all  points  equally  distant  from  two 
parallel  lines  ? 

9.  Prove  that  of  the  dihedral  angles  formed  by  a  plane  inter- 
secting parallel  planes,  the  alternate  and  corresponding  angles 
are  equal,  and  the  interior  angles  on  the  same  side  of  the  trans- 
versal plane  are  supplementary. 

TO.   Prove  that  all  plane  angles  of  a  dihedral  angle  are  equal. 


232  LINES  AND   PLANES   IN   SPACE      [VI,  §  271 

PART   IV.     POLYHEDRAL   ANGLES 

271.    Polyhedral  Angles.     The  figure  formed  by  three  ot 
more  straight  line  segments  that  end  in  a  common  point,  to- 
gether with  the  V-shaped  portions  of 
planes  determined  by  pairs  of  adjacent 
lines,  is  called  a  polyhedral  angle. 

The  point  at  which  the  lines  all  meet 
is  called  the  vertex  of  the  angle. 

The  lines  in  which  the  planes  meet 
are  its  edges ;  and  the  V-shaped  portions 
of  the  planes  between  these  edges  are 
its  faces.  Jq 

The  plane  angles  in  the  faces  at  the      yig.  183.    Polyhedral 
vertex  are  called  the  face  angles  of  the  Angle 

polyhedral  angle. 

A  polyhedral  angle  is  read  by  naming  the  vertex  and  a  point 
in  each  edge.  Thus,  in  Fig.  183,  the  polyhedral  angle  is  read 
V-ABCDE. 

Two  polyhedral  angles  are  congruent  if  they  can  be  placed 
so  that  their  vertices  coincide  and  their  corresponding  edges 
coincide. 

A  trihedral  angle  is  a  polyhedral  angle 
that  has  three  faces. 

Thus,  in  Fig.  184,  the  three  planes  VAB, 
VBC,  VAC,  which  meet  at  V  form  the 
trihedral  angle  V-ABC. 

Two  trihedral  angles  are  congruent  if  ^^^^b' 

the  three  face  angles  of  the  one  are  equal,     Fig.  184.    Trihedral 
respectively,  to  the  three  face  angles  of  Angle 

the  other,  and  are  arranged  in  the  same  order.     This  can  be 
shown  by  methods  similar  to  those  of  §  45.    See  also  §§  361, 374. 

If  the  intersections  of  a  plane  with  all  the  faces  of  a  poly- 
hedral angle  is  a  convex  polygon,  the  polyhedral  angle  is  a 
convex  polyhedral  angle. 


VT,  §  272]  POLYHEDRAL  ANGLES  233 

272.  Theorem  XII.     The  sum  of  tivo  face  angles  of  a 
trihedral  angle  is  greater  than  the  third. 

V 


Given  the  trihedral  angle  V-ABC. 

To  prove  t)i3.t  A  AVB -\- ABVG>  AAVC. 

Proof.  If  Z  AVGis  equal  to  or  less  than  either  of  the  other 
angles,  we  know  the  proposition  is  true  without  further  proof. 

If  Z.AVG  is  greater  than  either  of  the  other  angles,  lay  off 
any  lengths  VA  and  VG  on  the  sides  of  Z  AVG,  and  draw  AG. 
Then  draw  VD  in  the  plane  AVG,  making  /.AVD^^ZAVB. 

Lay  off  VB  =  VD,  and  draw  AB  and  GB. 

Then  AAVB^AA VD.  Why  ? 

Therefore  AB  =  AD.  Why  ? 

Kow  AB -{- BG  >  AD -{- DG.  Why? 

Whence,  subtracting,  BG>  DG.  Ax.  6 

Therefore  ZBVG>ZDVG.  §79 

By  construction        Z.  AVB  =  Z.  AVD. 

Adding,  .  Z  AVB -{- Z  BVG >  Z  AVG, 

EXERCISES 

1.  If  in  the  trihedral  angle  V-ABG,  Z  AVB  =  60°,  and 
Z  BVG  =  80°,  make  a  statement  as  to  the  number  of  degrees 
in  Z^  Fa 

2.  Show  that  any  face  angle  of  a  trihedral  angle  is  greater 
than  the  difference  of  the  other  two. 


234  LINES  AND   PLANES  IN  SPACE       [VI,  §  273 

273.  Theorem  XIII.  The  sum  of  the  face  angles  of 
any  convex  polyhedral  angle  is  less  than  four  right 
angles. 

V 


Fig.  186 

Given  the  polyhedral  angle  V-ABCDE  with  the  edges  cut 
by  any  plane  in  the  points  A,  B,  C,  D,  E. 

To  prove  that  the  sum  of  the  face  angles  of  the  polyhedral 
angle  is  less  than  four  right  angles. 

Proof.  Connect  any  point  0  in  the  polygon  ABODE  with 
the  vertices  A^  B,  O,  D,  E. 

The  number  of  triangles  with  the  common  vertex  O  is  the 
same  as  the  number  having  the  vertex  V. 

Now    Z  VBA  +  Z  VBOZABO  +  Z  OBC,  §272 

and  Z  VAB  +  Z  VAE  >  Z  BAO  +  Z  OAE,  etc. 

Therefore  the  sum  of  the  base  angles  of  the  triangles  having 
V  for  a  common  vertex  is  greater  than  the  sum  of  the  base 
angles  of  the  triangles  having  0  for  vertex. 

But  the  sum  of  all  the  angles  of  all  the  triangles  whose 
vertex  is  Fis  equal  to  the  sum  of  all  the  angles  of  all  the  tri- 
angles whose  vertex  is  0.  Why  ? 

Therefore  the  sum  of  the  angles  about  the  vertex  V  is  less 
than  the  sum  of  the  angles  about  0,  that  is,  less  than  four  right 
angles. 


VI,  §273]  MISCELLANEOUS  EXERCISES  235 

MISCELLANEOUS   EXERCISES   ON  CHAPTER  VI 

1.  Lean  one  book  in  a  slanting  position  against  another  book 
that  lies  flat  on  a  table,  and  hold  a  stretched  string  parallel  to 
the  cover  of  the  slanting  book.  Can  the  string  have  more  than 
one  position  ?     Can  the  string  be  horizontal  ?     Vertical  ? 

2.  Show  that  if  a  half-open  book  is  placed  on  a  table,  rest- 
ing on  its  bottom  edges,  the  back  edge  of  the  book  is  perpen- 
dicular to  the  plane  of  the  table,  and  the  lines  of  printing  are 
parallel  to  that  plane. 

3.  Show  that  the  dihedral  angle  between  the  pages  of  an 
open  book  is  measured  by  the  plane  angle  between  opposite 
lines  of  type  on  the  two  pages. 

4.  What  is  the  shape  of  the  end  of  an  ordinary  plank  after 
it  has  been  sawed  off  in  a  slanting  direction,  assuming  that  the 
opposite  faces  of  the  original  board  are  parallel  planes  ? 

5.  Prove  that  the  segments  of  two  parallel  lines  included  be- 
tween parallel  planes  are  equal. 

[Hint.  Pass  a  plane  through  the  parallel  lines  and  then  prove  that  the 
given  segments  form  the  opposite  sides  of  a  parallelogram. 

6.  Prove  that  a  plane  perpendicular  to  the  edge  of  a  dihe- 
dral angle  is  perpendicular  to  both  its  faces. 

[Hint.     Use  §  263.] 

7.  What  is  the  locus  of  all  the  points  equidistant  from  the 
three  faces  of  a  trihedral  angle  ? 

8.  Show  that  the  locus  of  any  given  point  on  a  line  seg- 
ment of  fixed  length,  whose  ends  touch  two  parallel  planes,  is 
a  third  plane  parallel  to  the  given  planes. 

9.  Prove  that  if  three  lines  are  perpendicular  to  each  other 
at  a  common  point  in  space,  each  line  is  perpendicular  to  the 
plane  of  the  other  two,  and  that  the  planes  of  the  lines  (taken 
in  pairs)  are  perpendicular  to  each  other.  Kote  how  this  is  il- 
lustrated on  a  cube,  or  in  a  corner  of  a  room,  or  in  a  corner  of 
an  ordiuary  box. 


236  LINES  AND   PLANES  IN  SPACE       [VI,  §  273 

10.  The  trihedral  angle  formed  when  three  planes  meet  each 
other,  so  that  each  is  perpendicular  to  the  other  two  is  called 
a  trirectangular  trihedral  angle. 

Prove  that  the  edges  of  a  trirectangular  trihedral  angle  are 
mutually  perpendicular  by  pairs.     See  §§  246,  265. 

11.  Prove  that  the  space  about  a  point  is  divided  into  eight 
congruent  trirectangular  trihedral  angles  by  three  planes  mutu- 
ally perpendicular  by  pairs  at  the  point. 

12.  Prove  that  if  a  line  is  parallel  to  the  intersection  of  two 
planes,  it  is  parallel  to  each  of  the  planes. 

[Hint.  Suppose  that  the  hne  is  not  parallel  to  one  of  the  planes  and 
thus  argue  to  an  absurdity.] 

13.  Prove  that  if  a  line  is  parallel  to  each  of  two  intersect- 
ing planes  it  is  parallel  to  their  intersection. 

14.  Prove  that  if  a  line  is  parallel  to  a  plane,  any  plane 
perpendicular  to  the  line  is  perpendicular  to  the  plane. 

[Hint.  Pass  a  plane  through  the  given  line  perpendicular  to  the  given 
plane  and  use  §  267.] 

15.  Can  a  trihedral  angle  be  formed  by  placing  three  equi- 
lateral triangles  so  that  one  vertex  of  each  lies  at  the  vertex 
of  the  trihedral  angle  ?         [Hint.    Use  §  273.] 

16.  Can  a  convex  polyhedral  angle  be  formed  as  in  Ex.  15 
by  placing  at  its  vertex  one  vertex  of  each  of  four  equilateral 
triangles  ?  Can  this  be  done  with  five  equilateral  triangles  ? 
With  six  ?     With  more  than  six  ? 

17.  Can  a  convex  polyhedral  angle  be  formed  by  placing  at 
its  vertex  one  vertex  of  each  of  three  squares  ?     Four  squares  ? 

18.  Can  a  convex  polyhedral  angle  be  formed  by  placing  at 
its  vertex  one  vertex  of  each  of  three  regular  pentagons  ? 
Four  ? 

19.  Show  that  just  five  different  convex  polyhedral  angles 
can  be  formed  as  in  Exs.  15-18  by  placing  at  a  single  point 
one  vertex  of  each  of  several  similar  regular  polygons. 


VI,  §  273]         MISCELLANEOUS  EXERCISES 


237 


M 


20.  Show  that  the  sum  of  the  dihedral  angles  of  a  trihedral 
angle  lies  between  two  and  six  right  angles. 

21.  Is  there  (in  general)  a  point  in  space  that  is  equidistant 
from  four  given  points  not  all  of  which  lie  in  the  same  plane  ? 
Give  reason  for  your  answer. 

22.  Given  any  line  I  and  a  plane  MN, 
drop  a  perpendicular  PA  from  any  point 
P  in  ?  to  MN.  Prove  that  I  and  PA  de- 
termine a  plane  perpendicular  to  MN. 
[This  plane  is  called  the  projecting  plane 
of  I  on  MN.  Its  intersection  AB  with  MN  is  called  the  pro- 
jection of  I  on  MN.  Define  similarly  the  projection  of  the 
segment  PQ.] 

23.  Prove  that  the  projection  on  a  plane  MN  of  the  line 
segment  joining  two  points  P  and  Q  (Ex.  22)  is  the  line  joining 
the  feet  A  and  jB  of  the  perpendiculars  dropped  to  the  plane 
from  P  and  Q,  respectively. 

24.  If  a  line  I  meets  a  plane  MN  at 
a  point  B,  prove  that  the  projection  of  I 
on  MN  is  the  line  joining  B  to  the  foot 
^  of  a  perpendicular  let  fall  from  any 
point  P  in  I.  [The  angle  ABP  between  the  line  I  and  its  pro- 
jection is  called  the  angle  between  the  line  and  the  plane j  or  the 
inclination  of  the  line  to  the  plane."] 

25.  Prove  that  the  sides  of  an  isosceles  triangle  make  equal 
angles  with  any  plane  containing  its  base. 

*26.  Show  that  the  length  of  the  projection  of  any  line 
segment  PQ  on  any  plane  is  the  length  of  PQ  times  the  cosine 
of  the  angle  between  the  line  and  the  plane. 


CHAPTER  VII 

POLYHEDRONS        CYLINDERS       CONES 

PART   I.     PRISMS 

274.  Polyhedrons.  A  polyhedron  is  a  limited  portion  of 
space  completely  bounded  by  planes.  The  intersections  of  the 
bounding  planes  are  called  the  edges ;  the  intersections  of  the 
edges,  the  vertices;  and  the  portions  of  the  bounding  planes 
bounded  by  the  edges,  the  faces,  of  the  polyhedron. 


ICOSAHJIDBOX        DODEGAHEDEON        OCTAHEDRON  HEXAHEDRON      TeTRAUEDROX 


A  polyhedron  of  four  faces  is  called  a  tetrahedron;  one  of 
six  faces  (for  example,  a  cube),  a  hexahedron;  one  of  eight 
faces,  an  octahedron;  one  of  twelve  faces,  a  dodecahedron;  one 
of  twenty  faces,  an  icosahedron. 

A  diagonal  of  a  polyhedron  is  a  straight  line  joining  any  two 
vertices  not  in  the  same  face. 

275.  Prisms.  A  prism  is  a  polyhedron,  two  of  whose  faces, 
called  its  bases,  are  congruent  polygons  in  parallel  planes,  and 
whose  other  faces,  called  lateral  faces,  are  parallelograms  whose 
vertices  all  lie  in  the  bases. 

A  triangular  prism  is  one  whose  base  is  a  triangle. 

The  sum  of  the  areas  of  the  lateral  faces  of  any  prism  is 
called  the  lateral  area  of  the  prism. 

238 


VII,  §  275] 


PRISMS 


239 


The  intersections  of  the  lateral  faces  are  the  lateral  edges 
of  the  prism. 

The  altitude  of  a  prism  is  the  perpendicular  distance  between 
its  bases. 


Right  Prisms 


Fig.  188 


Oblique  Prisms 


A  right  prism  is  one  whose  lateral  edges  are  perpendicular  to 
its  bases. 

An  oblique  prism  is  one  whose  lateral 
edges  are  oblique  to  its  bases. 

A  regular  prism  is  a  right  prism  whose 
bases  are  regular  polygons. 

Any  polygon  made  by  a  plane  which 
cuts  all  the  lateral  edges  of  a  prism,  as 
the  polygon  A'B'CD'E'  in  Fig.  189,  is 

called   a  section  of  the  prism.      A  right 

^.        .  J     1,  1  Fig.  189 

section  is  one  made  by  a  plane  perpen- 
dicular to  all  the  lateral  edges  of  the  prism,  as  ABODE. 


240  POLYHEDRONS  [VII,  §  276 

276.  Theorem  I.  The  sections  of  a  prism  made  by 
parallel  planes  cutting  all  the  lateral  edges  are  con- 
gruent polygons. 


Fig.  190 

Given  ABODE  and  AB'CD'M,  sections  of  the  prism  MN, 
made  by  parallel  planes. 

To  prove  that        ABODE  ^  A'B'  C'D'E'. 

Proof.     The  sides  of  the  polygon  ABODE  are  parallel  to  the 
sides  of  the  polygon  A!B'0'D'E'.  §  256 

Therefore  the  polygons  are  mutually  equilateral.  §  84 

Also  the  polygons  are  mutually  equiangular.  §  258 

Therefore  polygon  ABODE  ^  polygon  A'B'O'D'E',  §  33 

EXERCISES 

1.  How  many  edges  has  a  tetrahedron  ?     A  hexahedron  ? 

2.  How  many  diagonals  has  a  hexahedron  ?    A  tetrahedron  ? 

3.  Prove  that  any  two  lateral  edges  of  a  prism  are  equal  and 
parallel. 

4.  Prove  that  any  lateral  edge  of  a  right  prism  is  equal  to 
the  altitude. 

5.  Prove  that  all  right  sections  of  a  prism  are  congruent. 

6.  Prove  that  a  section  of  a  prism  parallel  to  the  base  is 
congruent  to  the  base. 


VII,  §  278]  PRISMS  241 

277.  Theorem  II.  The  lateral  area  A  of  a  prism  is 
equal  to  the  j^enineter  j^r  ^/  <^  right  section  multiplied 
hy  the  lateral  edge  e  ;  that  is,  A  —pr  x  6- 

J' 


Given  the  prism  ^D' with  a  right  section  PQIIST\  let  p^ 
denote  the  perimeter  of  the  right  section,  e  the  lateral  edge, 
and  A  the  lateral  area. 

To  prove   that  A=  Pr'X  ^• 

Proof.  The  lateral  area  consists  of  a  number  of  parallelo- 
grams, each  of  which  has  a  line  equal  to  A  A'  for  its  base.    Why  ? 

Each  of  these  parallelograms  has  one  of  the  sides  of  the 
right  section  PQRST  for  an  altitude.  Why  ? 

Therefore  the  areas  of  these  parallelograms  =  the  perimeter 
of  PQRS  T  multiplied  by  AA^.  Why  ? 

That  is  A=  Pr  y<  e. 

278.  Corollary  1.  The  lateral  area  A  of  a  right  prism  is  equal 
to  the  perimeter  of  its  base  multiplied  by  a  lateral  edge  ;  ov  A  = 
Pf,  X  e,  where  Pf,  denotes  the  perimeter  of  the  base,  and  e  de- 
notes a  lateral  edge. 

EXERCISES 

1.  Find  the  altitude  of  a  regular  prism,  one  side  of  whose 
triangular  base  is  5  in.  and  whose  lateral  area  is  195  sq.  in. 

2.  Show  that  the  lateral  area  of  a  regular  hexagonal  right 
prism  is  4V3  •  a/i,  where  h  is  the  altitude  and  a  the  distance 
from  the  center  of  the  base  to  one  of  the  sides. 


242  POLYHEDRONS  [VII,  §  279 

279.  Congruent  Solids.  Any  two  solids,  in  particular  any 
two  prisms,  are  said  to  be  congruent  when  they  can  be  made 
to  coincide  completely  in  all  their  parts. 

J  J' 


A, 

Congruent  Prisms 

280.  Theorem  III.  Two  prisms  are  congruent  if  three  faces 
that  include  a  trihedral  angle  of  one  are  congruent  respectively, 
and  similarly  placed,  to  three  faces  that  include  a  trihedral  angle 
of  the  other. 

Given  the  prisms  ^Jand  A  I'  with  face  ^J^  face  AJ\  face 
AO  ^  face  A'O',  and  face  AD  ^  face  A^D\ 

To  prove  that  prism  AI  ^  prism  J.' J'. 

Proof.  AEAF,  FAB,  and  EAB  are  equal  respectively  to 
A  E'A'F',  F'A'B',  and  E'A'B'.  Why  ? 

Then  trihedral  ZA^  trihedral  Z  A\  §  271 

Place  the  prism  AI  on  the  prism  A'l'  so  that  the  trihedral 
Z  A  coincides  with  its  congruent  trihedral  Z  A'. 

Then  the  face  AJ  will  coincide  with  the  congruent  face 
A'J';  AG  with  the  congruent  face  A'G';  and  AD  with  A'D' ; 
and  points  O  and  D  will  fall  on  G'  and  D'.  §  33 

Since  the  lateral  edges  of  a  prism  are  parallel  and  equal, 
CH  coincides  with  C'H',  and  DI  with  D'l'.  §§  257,  49 

Therefore  the  upper  bases  coincide,  and  the  prisms  coincide 
throughout  and  are  congruent. 

281.  Corollary  1.  Two  right  prisms  having  congruent  bases 
and  equal  altitudes  are  congruent. 


VII,  §  283] 


PRISMS 


243 


282.  Truncated  Prisms.  A  truncated  prism  is  a  portion  of 
a  prism  included  between  the  base  and  a  section  oblique  to  the 
base. 


Fig.  193  (a) 


Fig.  193  (6) 


283.  Corollary  2.  Two  truncated  prisms  are  congruent  if 
three  faces  including  a  trihedral  angle  of  the  one  are  congruent 
respectively  to  three  faces  including  a  trihedral  angle  of  the  other. 


EXERCISES 

1.  A  wooden  beam  10  ft.  long  has  a  rectangular  right  cross 
section  whose  dimensions  are  12  in.  by  16  in.  If  the  beam  be 
sawed  lengthwise  along  one  of  its  diagonal  planes,  show  that 
the  resulting  triangular  prisms  are  congruent. 

2.  What  will  be  the  lateral  area  of  one  of  the  triangular 
prisms  of  Ex.  1  ?    Its  total  area  ?      Ans.  40  sq.  ft. ;  411  sq.  ft. 

3.  A  carpenter  is  to  saw  from  a  given  square  piece  of  timber 
a  portion  of  which  one  end  is  to  be  perpendicular  to  the  lateral 
edges,  while  three  given  lateral  edges  are  to  be  3  ft.  6  in.,  3  ft. 
4  in.,  and  3  ft.  long,  respectively.  Show  that  these  measure- 
ments are  sufficient  to  enable  him  to  saw  off  the  desired  portion. 

4.  Show  that  to  make  a  right  prism  of  any  desired  shape,  it 
is  sufficient  to  have  a  pattern  of  a  right  section  of  the  desired 
prism,  and  the  length  of  one  lateral  edge. 

5.  Show  that  to  make  a  truncated  prism  of  any  desired 
shape,  of  which  one  end  is  a  right  section,  it  is  sufficient  to 
have  a  pattern  of  that  end,  and  the  lengths  of  three  given  con- 
secutive lateral  edges. 


244  POLYHEDRONS  [VII,  §  284 

284.  Theorem  IV.  An  oblique  prism  is  equal  in 
volume  to  a  right  prism  whose  base  is  a  right  section 
of  the  oblique  prism  and  whose  altitude  is  a  lateral 
edge  of  the  oblique  prism. 


Given  the  oblique  prism  AD' ;  and  given  a  right  prism  PS' 
whose  base  PS  is  a  right  section  of  the  prism  AD',  and  whose 
altitude  is  equal  to  a  lateral  edge  A  A'  of  the  prism  AD'. 

To  prove   that  prism  AD'  =  prism  PS'. 

Proof.  The  lateral  edges  of  the  prism  PS'  equal  the  lateral 
edges  of  the  prism  AD'.  Const. 

Therefore  AP  =  A'P',  BQ  =  B'Q',  etc.  Why  ? 

Moreover  PQ  =  P'Q',  and  the  face  angles  at  P,  Q,  P',  Q'  are 
right  angles.  Why  ? 

Hence,  by  superposition, 

Face  ^Q^  Face  .4' Q'. 
Likewise,  Face  BR  ^  Face  B'E',  etc. 

Now,  Section  PQBST  ^  Section  P'Q'R'S'T.      §  276 

Whence,  truncated  prism  AR  ^  truncated  prism  A'R'.  §  283 
Therefore,  truncated  prism  AR  +  truncated  prism  PD'  = 
truncated  prism  A'R'  4-  truncated  prism  PD'.  Ax.  1 

It  follows  that  prism  AD'  =  prism  PS'. 


VII,  §  286] 


PARALLELEPIPEDS 


245 


285.  Equivalent  Solids.  Two  solids  that  have  the  same 
volume  are  said  to  be  equivalent,  or  equal  in  volume. 

Thus  we  proved,  in  §  284,  that  any  oblique  prism  is  equiva- 
lent to  a  right  prism  whose  base  is  a  right  section  of  the  oblique 
prism  and  whose  altitude  is  equal  to  the  lateral  edge  of  the 
oblique  prism. 

286.  Parallelepipeds.  A  parallelepiped  is  a  prism  whose 
bases  are  parallelograms. 

A  right  parallelepiped  is  a  parallelepiped  whose  lateral  edges 
are  perpendicular  to  its  bases. 


Rectangular 
Parallelepiped 


Cube 


Oblique 
Parallelepiped 


Fig.  195 

A  rectangular  parallelepiped  is  a  right  parallelepiped  whose 
bases  are  rectangles. 

A  cube  is  a  parallelepiped  whose  six  faces  are  squares. 

An  oblique  parallelepiped  is  one  whose  lateral  edges  are 
oblique  to  its  bases. 

EXERCISES 

1.  Show  that  the  lateral  faces  of  a  right  parallelepiped  are 
rectangles.  ' 

2.  Find  the  sum  of  all  the  face  angles  of  a  parallelepiped. 

3.  Find  the  diagonal  of  a  cube  whose  edge  is  4  in. ;  20  in. ;  a. 


246  POLYHEDRONS  [VII,  §  287         \ 

287.  Theorem  V.  The  plane  passed  through  two  diagonally 
opposite  edges  of  a  parallelepiped  divides  the  parallelepiped  into 
two  triangular  prisms  that  are  equal  in  volume. 

G 


Given  the  plane  ACOE  passing  through  the  opposite  edges 
AE  and  CO  of  the  parallelepiped  AG. 

To  prove  that  the  parallelepiped  AG  is  divided  into  two 
equal  triangular  prisms,  ABG-F  and  ADC-H. 

Proof.     Let  IJKL  be  a  right  section  of  the  parallelepiped. 

From  the  definition  of  parallelepiped,  the  opposite  faces,  AF 
and  DG,  and  AH  and  BG,  are  parallel  and  congruent.    §  §  258, 88 

Therefore  IJ  II  LK,  and  IL  II  JK.  §  256 

Then  IJKL  is  a  parallelogram.  Why  ? 

The  intersection  IK  of  the  right  section  with  the  plane 
ACGE  is  the  diagonal  of  the  O  IJKL. 

Therefore  A IJK^  A IKL.  Why  ? 

But  the  prism  ABC-F\s  equal  in  volume  to  the  right  prism 
whose  base  is  Z/JS'and  altitude  AE,  and  the  prism  ACD-His 
equal  in  volume  to  the  right  prism  whose  base  is  ILK  and  al- 
titude AE.  §  284 

But  since  these  right  prisms  are  congruent,  §  281 

it  follows  that  prism  ABC-F =ipvism  ADC-H. 

Note.  If  the  faces  EFQH  and  ABCD  (Fig.  196)  are  perpendicular 
to  the  edges  AE,  BF,  etc.,  it  is  easy  to  see  that  the  diagonal  plane  AECG 
divides  the  parallelepiped  into  two  congruent  triangular  prisms.  This 
happens  for  any  rectangular  parallelepiped. 


VII,  §  287]  PARALLELEPIPEDS  247 

EXERCISES 

1.  Prove  tliat  the  diagonals  of  a  rectangular  parallelepiped 
are  equal,  and  that  the  square  of  the  diagonal  is  equal  to  the 
sum  of  the  squares  of  the  three  edges  that  meet  in  any  vertex. 

[Hint.  The  diagonal  is  the  hypotenuse  of  a  right  triangle  whose  sides 
are  one  of  the  edges  and  a  diagonal  of  one  face  ;  the  diagonal  of  the  face 
is  the  hypotenuse  of  a  right  triangle  whose  sides  are  two  of  the  edges.  ] 

2.  Find  the  length  of  the  diagonal  of  a  rectangular  parallel- 
epiped whose  edges  are  8,  10,  12. 

3.  Find  the  edge  of  a  cube  whose  diagonal  is  64  in. 

4.  Prove  that  the  diagonals  of  a  parallelepiped  bisect  each 
other. 

[Hint.  Consider  each  pair  of  diagonals  separately,  and  apply  §  87  to 
the  diagonal  plane  in  which  they  lie.] 

6.  Prove  that  if  the  right  section  IJKL  (Fig.  196)  of  a  par- 
allelepiped is  a  rectangle,  the  two  diagonal  planes  ACGE  and 
BDHF  divide  the  parallelepiped  into  four  triangular  prisms 
that  are  equal  in  volume. 

6.  A  tank  in  the  form  of  a  rectangular  parallelepiped  that 
holds  100  gal.  is  divided  into  four  compartments  by  two  vertical 
diagonal  planes.     What  is  the  capacity  of  each  compartment  ? 

7.  A  cube  each  of  whose  edges  is  1  ft.  long  is  called  a  unit 
cube;  its  volume  is  one  cubic  foot.  If  six  such  cubes  are 
placed  side  by  side  in  two  rows  of  three  each,  they  form  a  rec- 
tangular parallelepiped  2  ft.  wide,  1  ft.  high,  and  3  ft.  long. 
What  is  the  volume  of  this  parallelepiped  ?  What  is  the  vol- 
ume of  each  of  the  triangular  prisms  into  which  it  is  divided 
by  a  diagonal  plane  ? 

8.  How  many  unit  cubes  are  there  in  a  cube  each  of  whose 
edges  is  5  units  long  ? 

9.  How  many  unit  cubes  are  there  in  a  rectangular  par- 
allelepiped 3  units  long,  4  units  wide,  and  2  units  high  ?  What 
is  the  volume  of  this  parallelepiped  ? 


248 


POLYHEDRONS 


[VII,  §  288 


the 


^   ^1  /\  A 

y 

y" 

T  y\   y\   A  1 

^y<  y\    y\   /'\  Y^ 

<  y\   y\  A\^^ 

y\  y\    ^    > 

J- 

> 

y 

y 

y 

y 

y 

2 

y 

L 

B 


288.  Volume  of  a  Rectangular  Parallelepiped.  The  three 
edges  of  a  rectangular  parallelepiped  which  meet  at  a  common 
point  are  called  its  dimensions. 

In  Chapter  IV  (§  181),  we  assumed 
(without  proof)  the  well-known  principle 
that  the  area  of  a  rectangle  is  equal  to 
the  product  of  its  two  dimensions.  Simi- 
larly, we  shall  now  assume  that  the  vol- 
ume of  a  rectangular  parallelepiped  is 
equal  to  the  product  of  its  three  dimensions,  that  is,  to  the 
product  of  its  length,  breadth,  and  height ;  i.e. 

For  any  rectangular  parallelepiped  the  volume  V  is 

V=lxbxh, 

where   I,   h,   h  denote   the   length,   breadth,   and  height  of 
parallelepiped. 

The  student  is  reminded  that 
the  meaning  of  the  principle  is 
that,  if  the  three  dimensions  are 
each  measured  in  terms  of  the  same 
unit  of  length,  then  the  volume 
in  terms  of  the  corresponding  unit 
cube  is  the  product  of  the  three 
dimensions. 

The  following  corollaries  result  at  once  from  this  principle : 

289.  Corollary  1.     Two  rectangular  parallelepipeds  having 
congruent  bases  are  to  each  other 
as  their  altitudes. 

[Hint.  If  Z,  6,  and  h  represent 
the  dimensions  of  the  one  parallele- 
piped, then  Z,  6,  and  h'  will  represent 
the  dimensions  of  the  other.  The  cor- 
responding volumes  will  therefore  be 

to  each  other  in  the  ratio  {lbh)/{lbh')y  I  I 

that  is,  in  the  ratio  h/h'.^  Fia.  199 


& 


D 
Fig.  198 


/ 

^       / 

/     / 

h' 

h 

/       / 

VII,  §  292]  PARALLELEPIPEDS  249 

290.  Corollary  2.  Two  rectangular  parallelepipeds  having 
equal  altitudes  are  to  each  other  as  their  bases. 

291.  Corollary  3.  The  volume  of  a  cube  is  equal  to  the  cube 
of  its  edge. 

292.  Corollary  4.  The  volume  V  of  any  rectangular  parallele- 
piped is  the  product  of  the  area  of  its  base  B  and  its  altitude  h; 
that  is  J  V=Bxh. 

EXERCISES 

1.  Two  rectangular  parallelepipeds  with  equal  altitudes  have 
bases  containing  10  sq.  in.  and  15  sq.  in.,  respectively.  The 
volume  of  the  first  is  56  cu.  ft.  Find  the  volume  of  the 
second.  Ans.  84  cu.  ft. 

2.  Compare  the  volume  of  the  rectangular  parallelepiped 
whose  dimensions  are  8  in.,  10  in.,  11  in.  with  the  one  whose 
dimensions  are  1  ft.,  1  ft.,  and  16  in. 

3.  In  a  lot  120  ft.  long  and  66  ft.  wide  a  cellar  is  to  be 
dug  for  a  building.  The  cellar  is  to  be  44  ft.  long,  36  ft.  wide, 
and  7  ft.  deep.  The  earth  removed  is  to  be  used  to  fill  the 
surrounding  yard.     What  depth  of  fill  can  be  made  ? 

4.  A  standard  (U.S.)  gallon  contains  231  cu.  in.  How  many 
gallons  can  be  put  in  a  tank  of  the  form  of  a  rectangular  par- 
allelepiped that  is  2  ft.  high,  l^  ft.  wide,  and  3  ft.  long  ? 

5.  How  many  gallons  (see  Ex.  4)  are  there  in  1  cu.  ft.  ? 

6.  Find  the  size  of  a  cubical  tank  that  will  contain  50  gal. 

7.  Find  the  edge  of  a  cube  whose  volume  is  1728  cu.  in. ; 
of  a  cube  whose  volume  is  1500  cu.  in. 

8.  Find  the  diagonal  of  a  cube  whose  volume  is  521  cu.  in. 

9.  If  the  volume  of  one  cube  is  twice  that  of  another,  how 
do  their  edges  compare  ?  Ans.  V  2  : 1. 

10.  Find  the  edge  of  a  cube  whose  total  surface  is  60  sq.  ft. 

11.  The  edge  of  a  cube  is  a.  Find  the  area  of  a  section 
made  by  a  plane  through  two  diagonally  opposite  edges. 


250 


POLYHEDRONS 


IVII,  §  293 


293.  Theorem  VI.  The  volume  V  of  any  parallelepiped  is 
equal  to  the  product  of  its  base  B  and  its  altitude  h;  that  is, 
V=Bxh. 


Fig.  200 


Given  the  parallelepiped  /  with  its  volume  denoted  by  V, 
its  base  by  B,  and  its  altitude  by  h. 

To  prove  that  V=B  x  h. 

Proof.  Produce  AO  and  all  the  other  edges  of  /  that  are 
parallel  to  AC. 

On  the  prolongation  oi  AG  take  BE  =  AG,  and  through  D 
and  E  pass  planes  perpendicular  to  AE,  forming  the  right 
parallelepiped  II  whose  base  is  B'. 

Then  /=//.  §  284 

Prolong  FE  and  all  the  other  edges  of  II  that  are  parallel 
toi^^. 

On  the  prolongation  of  FE  take  MN  =  FE,  and  through  M 
and  N  pass  planes  perpendicular  to  FN,  forming  the  rec- 
tangular parallelepiped  III  whose  base  is  B". 

Then  II  =111  Why? 

Therefore  /=///.  Why? 


Moreover 


B'  =  B" 


Why? 


and  h,  the  altitude  of  7,  is  equal  to  the  altitude  of  III.     Why  ? 
But  the  volume  of  ///  is  B"  x  7i,  by  §  288;  hence  the  vol- 
ume of  /is  V=B''  xh  =  B  X  h. 


VII,  §  296] 


VOLUME  FORMULAS 


251 


294.  Theorem  VII.  The  volume  V  of  any  triangular  prism 
is  equal  to  the  product  of  its  base  B  and  its  altitude  h;  that 
is,V=Bxh.  ^r 

L;<=r- — ^~~■::::^-/' 

W      i 


M 
Fig.  201 

Given  the  triangular  prism  LMN-N'  whose   base  B  is  the 
triangle  LMN,  and  whose  altitude  is  li. 

To  prove  that  the  volume  V  of  LMN-N'  =  B  X  h. 

Proof.     Complete  the  parallelepiped  LMPN-P\ 

[The  remainder  of  the  proof  is  left  to  the  student.    Use  §  293.] 

295.   Corollary  1.     TJie  volume  V  of  any  prism  is  equal  to 
the  product  of  its  base  B  and  its  altitude  h;  that  is,  V—Bxh. 


Fig.  202 
[Hint.     Any  prism  may  be  divided  into  triangular  prisms  by  diagonal 
planes.] 

296.    Corollary  2.     Prisms  having  equivalent  bases  and  equal 
altitudes  are  equal. 


252  POLYHEDRONS  [VII,  §  296 

EXERCISES 

1.  Describe  one  or  more  ways  in  which  a  parallelepiped 
may  be  distorted  and  yet  have  its  volume  remain  unchanged. 

2.  The  base  of  a  parallelepiped  is  a  rhombus  one  of  whose 
diagonals  is  equal  to  its  side.  The  altitude  of  the  parallele- 
piped is  a,  and  is  also  equal  to  a  side  of  the  base.  Find  the 
volume  of  the  parallelepiped.  Ans.  a?  V3/2. 

3.  The  altitude  of  a  parallelepiped  is  3  in.,  and  a  diagonal 
of  a  base  divides  the  base  into  two  equilateral  triangles,  each 
side  of  which  is  6  in.     Find  the  volume  of  the  parallelepiped. 

4.  The  volume  of  a  rectangular  parallelepiped  is  2430  cu.  in. 
and  its  edges  are  in  the  ratio  of  3,  5,  and  6.     Find  its  edges. 

5.  The  altitude  of  a  prism  is  6  in.  and  its  base  is  a  square 
each  side  of  which  is  3  in.     Find  its  volume. 

6.  Show  that  two  prisms  with  equal  bases  are  to  each  other 
as  their  altitudes  j  and  that  those  with  equal  altitudes  are  to 
each  other  as  their  bases. 

7.  A  clay  cube  having  a  2-in.  edge  is  molded  into  the  form 
of  a  triangular  prism  of  height  3  in.  What  is  the  area  of  its 
base  ?  Does  it  make  a  difference  in  the  answer  whether  the 
prism  is  made  right  or  oblique  ?     Explain. 

8.  Assuming  that  iron  weighs  about  450  lb.  per  cu.  ft.,  find 
the  weight  of  a  rod  3  ft.  long,  whose  cross  section  is  a  rectangle 
11  by  2  in. 

9.  With  the  data  of  Ex.  8,  find  the  weight  of  an  iron  rod  2  ft. 
6  in.  long,  whose  cross  section  is  a  regular  hexagon  1  in.  on 
each  side. 

10.  What  must  be  the  length  of  the  side  of  an  equilateral 
triangle  in  order  that  a  triangular  prism  erected  upon  it  and  of 
height  1  ft.  shall  have  a  volume  of  1  cu.  ft.  ?  Solve  the  same 
problem,  when  it  is  assumed  that  the  base  is  a  regular  hexagon. 


VII,  §  297] 


PYRAMIDS 


253 


PART   II.     PYRAMIDS 

297.  Pyramids.  A  pyramid  is  a  polyhedron  bounded  by  a 
polygon,  called  its  base,  and  several  triangles  that  have  a 
common  vertex. 

The  triangles  are  called  the  lateral  faces,  the  common  vertex 
is  called  the  vertex  of  the  pyramid,  and  the  perpendicular  dis- 
tance from  the  vertex  to  the  base  is  called  the  altitude. 


Fig.  203.    Pyramid 

A  pyramid  is  triangular,  quadrangular,  etc.,  according  as  its 
base  is  a  triangle,  a  quadrilateral,  etc. 

A  regular  pyramid  is  one  whose  base  is  a  regular  polygon  and 
whose  vertex  lies  in  the  perpendicular  erected  at  the  center  of 
the  base. 

V 


B  C 

Fia.  204.    Regular  Pyramid 


The  slant  height  of  a  regular  pyramid  is  the  altitude  of  any 
one  of  its  triangular  faces  {VH in  Fig.  204). 


254 


POLYHEDRONS 


[VII,  §  297 


A  truncated  pyramid  is  the  portion  of  a  pyramid  included 
between  the  base  and  any  section  made  by  a  plane  cutting  all 
the  lateral  edges. 

A  frustum  of  a  pyramid  is  a  portion  included  between  the 
base  and  a  section  made  by  a  plane  parallel  to  the  base. 


Fig.  205.    Fkustum  of  a  Pyramid 

The  altitude  of  a  frustum  is  the  length  of  the  perpendicular 
between  the  planes  of  its  bases. 

The  lateral  faces  of  a  frustum  of  a  regular  pyramid  are  con- 
gruent trapezoids. 

The  slant  height  of  the  frustum  of  a  regular  pyramid  is  the 
altitude  of  one  of  the  trapezoids  forming  its  faces. 


EXERCISES 

1.  Of  which  type  are  the  celebrated  Egyptian  pyramids  ? 

2.  Prove  the  equality  of  the  lateral  edges  of  a  regular 
pyramid.     Of  those  of  a  frustum  of  a  regular  pyramid. 

3.  Prove  that  the  faces  of  any  frustum  of  a  pyramid  are 
trapezoids. 

4.  Prove  the  statement  made  in  §  297  that  the  faces  of  a 
frustum  of  a  regular  pyramid  are  congruent  trapezoids. 

5.  Prove  that  the  lateral  faces  of  a  regular  pyramid  are  con- 
gruent isosceles  triangles ;  hence  show  that  the  slant  height, 
measured  on  any  face,  is  the  same  as  that  measured  on  any  other 
face. 

6.  Prove  that  any  triangular  pyramid  is  a  tetrahedron. 
State  and  prove  the  converse. 


VII,  §  299] 


PYRAMIDS 


255 


298.  Theorem  VIII.  The  lateral  area  A  of  a  regular  pyra- 
mid is  equal  to  one  half  the  product  of  the  perimeter  of  its  base 
p,  and  its  slant  height  I  ;  that  is,  A=p  x  1/2. 

V 


Given  the  regular  pyramid  V-ABGDE  with  the  slant  height 
I,  the  lateral  area  A,  and  the  perimeter  of  the  base  p. 
To  prove  that  A=px  1/2. 

Proof.     A  =  the   sum   of   the  areas  of  the  triangles   VAB, 


VBC,  etc. 
Hence 


A  =  [AB  +  BC-^'-]xl/2=pxl/2. 


Why? 


299.  Corollary  1.  The  lateral  area  of  the  frustum  of  a 
regular  pyramid  is  equal  to  one  half  the  product  of  the  sum  of 
the  perimeters  of  the  bases  and  its  slant  height.     [See  §  191.] 

EXERCISES 

1.  The  slant  height  of  a  regular  hexagonal  pyramid  is  10  ft. 
Each  side  of  its  base  is  8  ft.  What  is  its  lateral  area  ?  Also, 
what  is  its  total  area  ?  Ans.  240  sq.  ft. ;  406.27  sq.  ft. 

2.  The  altitude  of  a  regular  quadrangular  pyramid  is  4  in. 
One  side  of  its  base  is  6  in.  What  is  its  lateral  area  ?  What 
is  its  total  area  ?  Ans.  60  sq.  in. ;  96  sq.  in. 

3.  Find  the  lateral  area  of  the  frustum  formed  by  a  plane 
bisecting  the  altitude  of  the  pyramid  mentioned  in  Ex.  2, 
Find  its  total  area. 


256 


POLYHEDRONS 


[VII,  §  300 


300.  Theorem  IX.    If  a  pyramid  is  cut  hy  a  plane  par- 

allel  to  the  base, 

(a)    The  altitude  and  the  lateral  edges  are  divided 
2?roportionally ; 

(h)  The  section  is  a  polygon  shnilaj'  to  the  base, 

V 


Given   the  pyramid  V-ABCDE  cut  by  a  plane  A^D'  parallel 
to  the  base  AD. 

To  prove  (a)  that 

VS/VS'  =  VA/  FA'  =  VB/  VB'  =  ...,  etc. 

(b)  that  the  section  A'B'CD'E'  is  similar  to  the  base. 

Proof  of  (a) 
Pass  a  plane  through  V  parallel  to  the  base  and  apply  §  261. 

Proof  of  (h) 
A  VAB  ~  A  VA'B' ;  A  VBC  ~  A  VB'C,  etc.     Why  ? 

Therefore 
AB/A'B'  =  VB/  VB'  j   VB/  VB'  =  BC/B'C,  etc. ;      Why  ? 
and  hence  AB/AB'  =  BC/B'C  =  CD /CD'  =  •  •  •  Ax.  9 

Thus,  the  polygons  ABODE  and  A'B'C'D'E'  have  their  cor- 
responding sides  proportional. 

Moreover,  the  same  polygons  are  mutually  equiangular.   §  258 

Hence  ABCDE  ^  A'B'C'D'E'.  §165 


Vn,  §  302]  PYRAMIDS  257 

301.  Corollary  1.  Parallel  sections  of  a  pyramid  are  to  each 
other  as  the  squares  of  their  distances  from  the  vertex. 

Proof :  In  Fig.  207,  ABCDE/A'B'C'D' E'  =  AB^/AJW\  §§  195,  300 

But                              AB/A'B'  =VB/VB',  Why? 

and  also                          VS/  VS'    =  VB/  VB' ;  (a) ,  §  300 

hence                              AB/A'B'  =  VS/VS'.  Ax.  9 

Whence,  squaring,       AB^/'AB'^  =  V^/VS^: 

Hence     ABCDE/A'B'C'D'E'  =  T^Vl^- 

302.  Corollary  2.  If  tivo  pyramids  that  have  equivalent  bases 
and  equal  altitudes  are  cut  by  planes  parallel  to  their  bases  and 
at  equal  distances  from  their  vertices,  the  sections  are  equivaleyit. 

[Hint.  Represent  by  B  and  B'  the  areas  of  the  two  sections,  and  by  B 
and  B'  the  areas  of  the  bases.  Let  h  be  the  common  altitude  of  the  pyra- 
mids, and  k  the  distance  from  the  vertex  of  either  pyramid  to  the  section 
made  in  it.  Then  B/B  =  k^h'^  and  B'/B'  =  k^/h'^  (§  301)  ;  hence  B/B 
=  B'/B'.    But  B  =  B'  hy  hypothesis  ;  hence  B  =  B'.} 

EXERCISES 

1.  Compare  the  areas  of  two  sections  of  a  pyramid  whose 
perpendicular  distances  from  the  vertex  are  3  in.  and  4  in. 
respectively.  Does  it  make  any  difference  in  your  answer 
whether  the  pyramid  is  of  one  shape  or  another  ?     Ans.  9  :  16. 

2.  The  altitude  of  a  pyramid  with  a  square  base  is  16  in., 
the  area  of  a  section  parallel  to  the  base  and  10  in.  from  the 
vertex  is  5Q\  sq.  in.     Find  the  area  of  the  base. 

3.  The  bases  of  the  frustum  of  a  regular  pyramid  are  equi- 
lateral triangles  whose  sides  are  10  in.  and  18  in.  respectively ; 
the  altitude  of  the  frustum  is  8  in.  Find  the  altitude  of  the 
pyramid  of  which  the  given  figure  is  a  frustum.        Ans.  18  in. 

4.  The  altitude  of  a  pyramid  is  H.  At  what  distance  from 
the  vertex  must  a  plane  be  passed  parallel  to  the  base  so  that 
the  section  formed  shall  be  (1)  one  half  as  large  as  the  base  ? 
(2)  one  third  ?     (3)  one  ninth  ? 


258 


POLYHEDRONS 


[VII,  §  303 


303.  Theorem  X.      Tioo  triangular  pyramids  having 
eguivalent  bases  and  equal  altitudes  are  equivalent. 


Fig.  208 

Given  the  pyramids  V-ABC  and  V'-A'B'C  having  equiva- 
lent bases  ABC,  A'B'C,  and  a  common  altitude  AH. 

To  prove  V-ABC  =  V'-A'B'C. 

Proof.  The  proof  of  the  theorem  consists  in  showing  that^ 
the  pyramids  V-ABC  and  V'-A'B'C  cannot  differ  in  volume 
by  as  much  as  any  given  amount,  however  small.  This  means 
that  the  two  volumes  are  actually  equal,  for  if  they  were 
unequal,  they  would  differ  by  as  much  as  some  Jixed  amount,  — 
in  fact,  that  is  what  unequal  means. 

We  proceed,  then,  to  show  that  V-ABC  and  V'-A'B'C^  can- 
not differ  by  as  much  as  any  given  amount,  however  small. 

Divide  the  altitude  AH  into  a  number  of  equal  parts,  and 
through  the  points  of  division  pass  planes  parallel  to  the  plane 
of  the  bases. 

The  corresponding  sections  made  by  any  one  of  these  planes 
in  the  two  pyramids  are  equivalent.  §  302 

Now  inscribe  in  each  pyramid  a  series  of  prisms  having  the 
triangular  sections  as  upper  bases  and  the  distance  between 
the  sections  as  their  common  altitude. 


VII,  §304]  PYRAMIDS  259 

Each  pair  of  corresponding  prisms  in  the  two  pyramids  are 
then  equivalent.  §  296 

Therefore,  the  sum  of  the  prisms  inscribed  in  V-ABG  is 
equivalent  to  the  sum  of  the  prisms  inscribed  in  V-A'B'C 

If  the  number  of  divisions  into  which  AH  is  divided  is  taken 
sufficiently  large,  the  sum  of  the  prisms  in  V-ABG  may  be 
made  to  differ  from  the  volume  of  V-ABG  by  less  than  any 
given  amount.  Likewise,  by  taking  the  number  of  divisions 
in  AH  sufficiently  large,  the  sum  of  the  prisms  in  V'-A'B'O 
may  be  made  to  differ  from  the  volume  of  V'-A!B'G'  by  less 
than  the  same  given  amount,  however  small  it  has  been  taken. 

Since,  by  taking  the  number  of  divisions  in  AH  sufficiently 
large,  the  volumes  of  V-ABG  and  V'-A'B'G'  differ  by  less 
than  any  given  amount  from  these  equal  sums,  the  pyramids 
must  differ  from  each  other  by  less  than  the  same  given  amount, 
and  this  is  what  we  were  to  show.     Compare  §  211. 

304.  Corollary  1.  Ayiy  two  pyramids  having  equivalent  bases 
and  equal  altitudes  are  equivalent. 


Fig.  209 
[Hint.     Divide  each  pyramid  into  triangular  pyramids.] 

EXERCISES 

1.  What  is  the  locus  of  the  vertices  of  all  pyramids  whose 
bases  and  volumes  are  the  same  ? 

2.  Prove  that  if  the  base  of  a  pyramid  is  a  parallelogram,  the 
plane  determined  by  its  vertex  and  either  diagonal  of  its  base 
divides  it  into  two  equivalent  triangular  pyramids. 


260 


POLYHEDRONS 


[VII,  §  305 


305.  Theorem  XI.  The  volume  V  of  a  triangular  pyramid 
is  equal  to  one  third  the  product  of  its  base  B,  and  its  altitude 
h;  that  is,  V=Bxh/3. 


Fig.  210 

Given  the  triangular  pyramid  0-LMN. 

To  prove  that  the  volume  V  of  0-LMN  equals  \  its  base  B 
times  its  altitude  }i ;  that  is,  V  =  5  X  /i/3. 

Proof.  Construct  a  triangular  prism  MP  having  LMN  iov 
its  base,  and  its  lateral  edges  equal  and  parallel  to  the  edge  OM. 

The  prism  MP  is  made  up  of  the  triangular  pyramid 
0-LMN  and  the  quadrangular  pyramid  0-LNPQ. 

Pass  a  plane  through  OQ  and  ON  dividing  the  quadran- 
gular  pyramid   into   two   triangular  pyramids,   0-LNQ   and 

0-NQP. 

Pyramid  0-LNQ  =  pyramid  0-NQP.  §  303 

Pyramid  0-NQP  may  be  read  N-QOP. 
Pyramid  N-QOP  =  pyramid  0-LMN  §  303 

Therefore,  the  three  triangular  pyramids  are  equal  and 
0-LMN  is  one  third  the  prism. 

But  the  volume  of  the  prism  is  equal  to  the  product  of  its 
base  and  its  altitude.  §  294 

Therefore,  pyramid  0-LMN  =  ^  the  product  of  its  base 
and  its  altitude. 


VII,  §  307] 


PYRAMIDS 


261 


306.  Corollary  1.  The  volume  V  of  any  pyramid  is  equal 
to  one  third  the  product  of  its  base  B  and  its  altitude  h;  that  is. 
V=Bh/3. 


Fig.  211 
[Hint.      Divide  the  pyramid  into  triangular  pyramids  and  apply  §  305. ] 

307.  Theorem  XII.  If  a  frustum  of  a  pyramid  has  bases 
B  a7id  B'  ayid  altitude  h,  and  is  cut  from  a  pyramid  P  whose  base 
is  B  and  whose  altitude  is  H,  the  volume  V  of  the  frustum  is 
given  by  the  formula  : 

y^BH     B'(H-h) 
3  3         * 

Given  the  pyramid  P  with  base  B  and 
altitude  H,  and  a  frustum  of  it  with  bases 
B  and  B'  and  altitude  h. 

To  prove  that  the  volume  Vof  the  frus- 
tum is 

V=  ^^     B'jH-h) 


'-     3               3 

Fig. 
Proof.     The  frustum  is    the  difference 

M 
212 

between  the  two  pyramids  P  and  P',  where  P'  has  tl 

le  base  B 

and  the  same  vertex  as  P. 

The  volume  of  P  is  BH/3. 

Why? 

Since  the  altitude  of  P'  is  H  —  h,  its  volume  is 

B'(H-h) 
3 

Why? 

Hence            V-P     P  =  ^^     ^'(^-^>. 

o                 o 

262 


POLYHEDRONS 


[VII,  §  308 


308.   Corollary  1.     The  volume  V  of  a  frustum  of  a  pyramid 
of  bases  B  and  B'  and  altitude  h  is  given  by  the  formula : 

V={B  +5'  +VBByi  /3. 

Outline  of  Proof.     By  §  301, 

using  the  notation  of  §  307. 

Hence    VB'/VB  =  {H-h)/n=l-  h/H, 
whence  H=h  Vb/{  VB  -  VW) . 

But,  by  §  307, 

V=  BH/S  -B'(H-  h)/S 
=  (B-  B')H/S  +  B'h/3. 
Substituting  the  value  of  H  just  found,  we 
have 

V  =  [(^-^')^  ^B'l h/S  =[B  +  VBB'  +  B'}h/S. 
^  VB-  y/B'  J 


EXERCISES 

1.  Find  the  altitude  of  a  triangular  pyramid  whose  volume 
is  50  cu.  in.  and  whose  base  is  12  sq.  in.  Ans.  12^  in. 

2.  If  a  prism  and  a  pyramid  have  a  common  base  and  alti- 
tude, what  is  the  ratio  of  their  volumes  ? 

3.  If  the  base  of  a  pyramid  is  a  square  and  its  altitude  is 
3  ft.,  how  long  must  each  side  of  the  square  be  in  order  that 
the  volume  may  be  16  cu.  ft.  ? 

4.  Show  that  the  volume  of  the  tetrahedron,  all  of  whose 
edges  are  equal  to  a,  is  V2  ayi2. 

[Hint.     See  Th.  XXXIII,  §  102.] 

5.  Find  the  volume  of  a  frustum  of  the  pyramid  of  Ex.  1 
cut  off  by  a  plane  6  in.  from  the  base. 

6.  Find  the  volume  of  each  of  the  parts  into  which  the  pyr- 
amid of  Ex.  3  is  cut  by  two  planes  parallel  to  its  base  which 
trisect  the  altitude. 


VII,  §  309] 


CYLINDERS 


263 


PART   III.     CYLINDERS    AND   CONES 


309.  Cylinders.  A  cylindrical  surface  is  a  curved  surface 
generated  by  a  moving  straight  line,  called  the  generatrix, 
which  moves  always  parallel  to  itself  and  constantly  passes 
through  a  fixed  curve  called  the  directrix.  The  generatrix  in 
any  one  position  is  called  an  element  of  the  surface.  One 
element  and  only  one  can  be  drawn  through  a  given  point  on 
the  cylindrical  surface. 

A  cylinder  is  a  solid  bounded  by  a  cylindrical  surface  and 
two  parallel  planes.  The  two  plane  surfaces  are  called  the 
bases,  and  the  cylindrical  surface  is  called  the  lateral  surface. 

The  altitude  of  a  cylinder  is  the  length  of  the  perpendicular 
between  the  bases. 

A  right  section  of  a  cylinder  is  a  section  made  by  a  plane 
perpendicular  to  all  its  elements. 


Cylindrical  Surface 


Right  Cylinder 
Fig.  214 


Oblique  Cylinder 


A  circular  cylinder  is  one  whose  bases  are  circles. 

A  right  cylinder  is  one  whose  elements  are  all  perpendicular 
to  its  bases ;  otherwise,  the  cylinder  is  said  to  be  oblique. 

A  right  circular  cylinder  is  a  right  cylinder  whose  base  is  a 
circle.     Such  a  cylinder  can  be  generated  by  the  revolution  of 


264 


CYLINDERS 


[VII,  §  310 


a  rectangle  about  one  of  its  sides  as  an  axis ;  for  this  reason  a 
right  circular  cylinder  is  sometimes  called  a  cylinder  of  revo- 
lution. 

310.  Postulate.  A  prism  is  inscribed  in  a  cylinder  when  its 
lateral  edges  are  elements  of  the  cylinder  and  its  bases  are 
inscribed  in  the  bases  of  the  cylinder. 


Fia.  215 

In  order  to  study  the  properties  of  the  cylinder  the  following 
postulate  is  needed : 

A  cylinder  and  a  prism  inscribed  within  it  may  he  made  to 
differ  by  as  little  as  we  please,  both  in  lateral  area  and  in  volumCj 
by  making  the  number  of  sides  of  the  base  of  the  prism  sufficiently 
great,  while  the  length  of  each  of  those  sides  becomes  sufficiently 
small. 

The  length  of  an  edge  of  the  inscribed  prism  is  equal  to  the 
length  of  an  element  of  the  cylinder  (see  Ex.  5,  p.  235) ;  and, 
by  increasing  the  number  of  sides  of  the  inscribed  prism, 
the  base  of  the  prism  approaches,  both  in  area  and  in  the 
length  of  its  perimeter,  as  nearly  as  we  please  to  the  base  of 
the  cylinder.     This  latter  fact  we  assume,  as  in  §  210. 

We  shall  now  proceed  to  show  that  the  theorems  already 
proved  for  prisms  can  be  extended  to  cylinders  by  the  use  of 
the  preceding  postulate. 


VII,  §  313] 


AREAS  AND   VOLUMES 


265 


311.  Theorem  XIII.  The  lateral  area  A  of  any  cylinder 
is  equal  to  the  product  of  an  element  I  and  the  perim- 
eter p  of  a  right  section  ;  that  is,  A  =  lxp. 


Fig.  216 

Outline  of  Proof.    In  order  to  prove  the  theorem,  inscribe  in 
the  cylinder  any  prism  whose  base  is  a  polygon  of  n  sides. 
Then,  for  this  pris7n,  by  §  277  : 
(1)  A'  =  lxp', 

where  A'  and  p'  are,  respectively,  the  Jateral  area  and  the  pe- 
rimeter of  the  right  section  of  the  prism ;  and  where  I  is  the 
length  of  an  edge  of  the  prism,  which  is  equal  to  an  element  of 
the  cylinder.  As  the  number  n  of  sides  increases  so  that 
the  length  of  each  side  approaches  zero, 

A'  comes  to  differ  from  A  by  as  little  as  we  please ;       §  310. 

Ixp'  comes  to  differ  from  Z  xp  by  as  little  as  we  please.  §  310. 
Hence,  by  (1),  A  comes  to  differ  from  I  xp  by  as  little  as  we  please. 

It  follows,  as  in  §  303,  that  A  =1  X  p. 

312.  Corollary  1.  TJie  lateral  area  of  a  right  circular 
cylinder  is  equal  to  2  nrh,  where  r  is  the  radius  of  the  circular 
base  and  h  is  the  altitude  of  the  cylinder.     See  §  214. 

313.  Corollary  2.  Tlie  lateral  area  of  any  cylinder  whose 
right  section  is  a  circle  is  equal  to  2  rrrrl,  where  r  is  the  radius  of 
the  right  section,  and  I  is  the  length  of  an  element. 


266  CYLINDERS  [VII,  §  314 

314.  Theorem  XIV.  The  volume  V  of  any  cylinder  is 
equal  to  the  product  of  its  base  B  and  its  altitude  h  ; 
that  is,  V=Bxh. 

[The  proof  is  left  to  the  student.  Inscribe  a  prism  of  n  sides  in  the  cyl- 
inder and  use  §  295  and  §  310.    Follow  the  steps  suggested  by  §  311.] 

315.  Corollary  1.  The  volume  of  a  circular  cylinder  is  equal 
to  TrrVi,  where  r  is  the  radius  of  the  base  and  h  is  the  altitude  of 
the  cylinder.     See  §  216. 

EXERCISES 

[In  these  exercises,  use  the  approximate  value  tt  =  22/7.  J 

1.  In  a  steam  engine  65  flues,  or  cylindrical  pipes,  each  2  in. 
in  outside  diameter  and  12  ft.  long,  convey  the  heat  from  the 
fire-box  through  to  the  water.  How  much  heating  surface  is 
presented  to  the  water  ?  Ans.  408^  sq.  ft. 

2.  Neglecting  the  lap,  how  much  tin  is  required  to  make  a 
stovepipe  10  ft.  long  and  8  in.  in  diameter  ? 

3.  A  right  circular  cylinder  has  the  radius  of  its  base  equal 
to  3  in.  How  great  must  its  altitude  be  in  order  that  it  shall 
have  a  lateral  area  of  30  sq.  in.  ? 

4.  Find  the  total  area,  including  the  ends,  of  a  covered  tin 
can  whose  diameter  is  4  in.  and  whose  height  is  6  in. 

Ans,  1004^  sq.  in. 

5.  Derive  a  general  formula  for  the  total  area  (including  the 
bases)  of  a  right  circular  cylinder  whose  height  is  h  and  the 
radius  of  whose  base  is  r. 

6.  What  fraction  of  the  metal  in  a  tin  can  5  in.  wide  and 
5  in.  high  is  used  to  make  the  top  and  bottom  ?  What  to  make 
the  circular  sides  ? 

7.  If  the  diameter  of  a  well  is  7  ft.  and  the  water  is  10  ft. 
deep,  how  many  gallons  of  water  are  there  in  it,  reckoning 
H  gal.  to  the  cubic  foot  ?  Ans.  2887.5  gals. 


VII,  §  315]  AREAS  AND  VOLUMES  267 

8.  When  a  body  is  placed  in  a  cylindrical  tumbler  of  water 
3  in.  in  diameter,  the  water  level  rises  1  in.  What  is  the 
volume  of  the  body?  Note  that  a  method  for  finding  the 
volume  of  a  body  of  any  shape  is  here  illustrated. 

9.  Show  that  the  volume  Fand  the  lateral  area  ^  of  a  right 
circular  cylinder  are  connected  by  the  relation  V=  Ax  r/2. 

10.  One  gallon  is  231  cubic  inches.  At  what  heights  on  a 
cylindrical  measuring  can  whose  base  is  6  in.  in  diameter  will 
the  marks  for  1  gallon,  1  quart,  2  quarts,  3  quarts,  be  made  ? 

11.  Find  the  total  area  of  the  gallon  measuring  can  of 
Ex.  10. 

12.  Having  given  the  total  surface  !r  of  a  right  circular  cylin- 
der, in  which  the  height  is  equal  to  the  diameter  of  the  base, 
find  the  volume  V- 

[Hint.  —Call  the  height  h.'] 

13.  Eind  the  volume  of  metal  per  foot  of  length  in  a  pipe 
whose  outer  diameter  is  3^  in.,  and  whose  inner  diameter  is 
3  in.  Hence  find  the  weight  per  foot  of  length  if  the  pipe 
is  iron,  it  being  given  that  iron  weighs  (about)  450  lb.  per 
cubic  foot.  ^^^TTJT'-^^ 

14.  If  the  outer  and  inner  diameters  of  a  /^^^^^^^ 
tube  are  D  and  d,  respectively,  show  that  the  '^M_  ^-^^-^^ 
volum^e  in  a  length  I  is  ttI{D'^  —  d2)/4.  If  the  HT'^  "^B 
thickness  of  the  tube  is  denoted  by  t,  show  '^^^^^^^^i 
that  t—(D  —  d)/2,  and  hence  that  the  volume  j---^^^^^fl_4 
m  a  length  I  is  i< d ^ 

'7Tlt{D  +  d)/2. 

15.  Show  that  the  volume  of  the  tube  of  Ex.  14  can  also  be 
represented  by  the  formula  Trlt{d  + 1)  ;  or  by  the  formula 
Trlt{D  —  t),  in  the  notation  of  Ex.  14. 

16.  What  is  the  volume  of  the  largest  beam  of  square  cross 
section  that  can  be  cut  from  a  circular  log  16  in.  in  diameter 
and  10  ft.  long  ?     What  fraction  of  the  log  is  wasted  ? 


268 


CONES 


[VII,  §  316 


316.  Cones.  A  conical  surface  is  a  surface  generated  by  a 
moving  straight  line  AB,  called  the  generatrix,  which  passes 
always  through  a  fixed  point  V,  called  the  vertex,  and  constantly 


Fig.  217 

intersects  a  fixed  curve,  called  the  directrix.  A  conical  surface 
thus  consists  of  two  parts,  which  are. called  the  upper  and 
lower  nappes.  The  generatrix  in  any  one  position  is  called  an 
element  of  the  surface. 

A  cone  is  a  solid  bounded  by  a  conical  surface  and  a  plane 
which  cuts  all  its  elements.  The  plane  is  then  called  the  base 
of  the  cone ;  and  the  conical  surface  is  called  the  lateral  surface 
of  the  cone.  The  altitude  of  a  cone  is  the  perpendicular  dis- 
tance from  its  vertex  to  its  base. 


Fig.  218 


A  circular  cone  is  one  whose  base  is  a  circle.    The  axis  of  a  cir- 
cular cone  is  the  line  joining  the  vertex  to  the  center  of  the  base. 


VII,  §  316] 


AREAS  AND  VOLUMES 


269 


A  right  circular  cone  is  a  circular  cone  whose  axis  is  perpen- 
dicular to  the  base.  Such  a  cone  is  also  called  a  cone  of  revolu- 
tion, since  it  may  be  generated  by  revolving  a  right  triangle 
about  one  of  its  sides  as  an  axis. 


LV  B  VM 

Fig.  219.    Right  Circular  Cone  and  Section  Parallel  to  Base 

The  slant  height  of  a  cone  of  revolution  is  the  length  of  one 
of  its  elements. 

A  frustum  of  a  cone  is  the  portion  of  a  cone  included  be- 
tween the  base  and  a  section  parallel  to  the  base  and  cutting 
all  the  elements. 


Fig.  220.    Frustum  of  a  Cone 

The  lateral  surface  of  a  frustum  of  a  cone  is  the  portion  of 
the  lateral  surface  of  the  cone  included  between  the  bases  of 
the  frustum. 

The  slant  height  of  a  frustum  of  a  cone  of  revolution  is  the 
portion  of  any  element  of  the  cone  included  between  the  bases. 


270 


CONES 


[VII,  §  317 


317.  Postulate.  A  pyramid  is  inscribed  in  a  cone  when  its 
lateral  edges  are  elements  of  the  cone  and  its  base  is  inscribed 
in  the  base  of  the  cone.  I 

The  following  postulate,  corresponding  to  that  of  §  310,  is 
needed  in  the  study  of  the  cone  : 

A  cone  and  pyramid  inscribed  within  it  may  be  made  to  differ 
by  as  little  as  lue  please,  both  in  lateral  area  and  in  volume,  by 
making  the  number  of  sides  of  the  pyramid  sufficiently  great,  while 
the  length  of  each  side  of  the  base  becomes  sufficiently  small. 


Fig.  221.    Cone  with  Inscribed  Pyramid 

The  base  of  the  inscribed  pyramid  approaches,  both  in  area 
and  in  perimeter,  the  base  of  the  cone  (§  310);  and  the  alti- 
tude of  any  face  of  the  pyramid  approaches  an  element  of  the 
cone,  as  the  pyramid  approaches  the  cone. 

If  the  cone  is  a  right  circular  cone,  the  pyramid  can  be  made 
a  regular  pyramid ;  then  the  slant  height  of  the  pyramid  ap- 
proaches the  slant  height  of  the  cone.     ' 

318.  Restriction.  The  word  cone  as  used  hereafter  in  this 
book  will  be  understood  to  refer  to  a  circular  cone  only.  The 
preceding  postulate  applies,  however,  to  any  kind  of  cone; 
and  it  may  be  used  to  obtain  results  for  cones  of  any  form  in 
the  manner  illustrated  below  for  circular  cones  only. 

We  proceed  to  extend  to  circular  cones  certain  theorems 
already  proved  for  pyramids. 


VII,  §  321] 


AREAS  AND  VOLUMES 


271 


319.-  Theorem  XV.  The  lateral  area  A  of  a  right 
circular  cone  is  equal  to  one  half  the  product  of  its 
slant  height  I  and  the  circumfere7ice  p  of  its  base; 
that  is,  A  =  lxp/2. 

Outline  of  Proof.  Inscribe  a  regular  pyramid  of  n  faces  in 
the  cone  (see  Fig.  221) ;  then,  by  §  298,  the  lateral  area  A'  of 
the  pyramid  is  equal  to  one  half  the  product  of  its  slant  height 
V  and  the  perimeter  p'  of  its  base ;  that  is, 

A'=irxp'', 

and   this  formula  is  correct  no  matter  how  large  n  may  be. 

By  taking  n  sufficiently  large,  A'  comes  to  differ  by  as  little 
as  we  please  from  A ;  while  Z'  and  p'  come  to  differ  by  as  little 
as  we  please  from  I  and  p,  respectively.  §  317. 

Whence,  as  in  §  311, 

A  =  ^lxp. 

320.  Corollary  1.  T7ie  lateral  area  of  a  right  circular  cone 
is  irr  '  I,  where  r  is  the  radius  of  the  base  and  I  is  the  slayit  height. 
See  §§  319  and  214. 

321.  Corollary  2.  The  lateral  area  of  a  frustum  of  a  right 
circular  cone  is  equal  to  one  half  the  product  of  its  slant  height 
and  the  sum  of  the  circumferences  of  its  bases. 


Fig.  222 


[The  proof  is  left  to  the  student.     Inscribe  a  frustum  of  a  regular 
pyramid  in  the  given  frustum  of  a  cone,  and  use  §  317  and  §  299.] 


272 


CONES 


[VII,  §  322 


322.  Theorem  XVI.  Any  section  of  a  circular  cone  paralleX 
to  its  base  is  a  circle  tvhose  area  is  to  that  of  the  base  as  the  squan 
of  its  distance  from  the  vertex  is  to  the  square  of  the  altitude  of 

the  cone.  [Hint.  To  prove  that  the  section  is  a  circle,  pass  any  two 
planes  through  the  axis  of  the  cylinder,  and  show  that  their  intersections 
with  the  section  are  equal.  Then  inscribe  a  regular  pyramid  and  proceed 
as  in  §  319,  using  §§  301  and  317.] 

323.  Theorem  XVII.  71ie  volume  V  of  a  cojie  is 
equal  to  one  third  the  product  of  its  base  B  and  its 
altitude  h  ;  that  is,  V  ==  Bh/S. 

[Hint.     Use  §§  317,  306,  and  proceed  as  in  §  319.] 

324.  Corollary  1.  Jf  from  any  cone  whose  base  is  B  and 
whose  altitude  is  H,  a  frustum  is  cut,  whose  upper  base  is  B'  and 
whose  altitude  is  h,  the  volume  V  of  the  frustum  is 


'V 

/h| 

\ 

1 

"'"'^p 

_[jn   \ 

^    1 

1  r\ 

Fig.  223 

325.  Corollary  2.  The  volume  of  a  frustum  of  any  cone  is 
equal  to  the  sum  of  three  cones  ivhose  common  altitude  is  the  alti- 
tude of  the  frustum  and  whose  bases  are  the  two  bases  and  a  mean 
proportional  between  them. 

[Hint.    Use  §§  322,  324,  noting  also  §  308.] 


VII,  §325]  AREAS  AND   VOLUMES  273 

EXERCISES 

1.  The  altitude  of  a  right  circular  cone  is  12  in.  and  the 
radius  of  the  base  9  in.  Find  the  lateral  area  and  the  total 
area  of  the  cone.  Ans.  424^^  sq.  in. ;  678f  sq.  in. 

2.  How  many  square  yards  of  canvas  are  there  in  a  conical 
tent  12  ft.  in  diameter  and  8  ft.  high  ? 

3.  The  total  area  of  a  right  circular  cone  whose  altitude  is 
10  in.  is  280  sq.  in.  Find  the  total  area  of  the  cone  cut  off 
by  a  plane  parallel  to  the  base  and  6  in.  from  the  vertex. 

4.  The  altitude  of  a  right  circular  cone  is  H.  How  far  from 
the  vertex  must  a  plane  be  passed  parallel  to  the  base  so  that 
the  lateral  area  of  the  cone  cut  off  shall  be  one  half  that 
of  the  original  cone  ?  Ans.  H/V2. 

[Hint.  First  prove  that  the  area  of  the  cross  section  made  by  the 
plane  will  be  one  half  the  area  of  the  base.     Then  apply  §  322.] 

5.  The  slant  height  and  the  diameter  of  the  base  of  a  right 
circular  cone  are  each  equal  to  /.  Find  the  total  area,  includ- 
ing the  base. 

6.  The  circumference  of  the  base  of  a  circular  cone  is  11  ft. 
and  its  height  is  8  ft.     What  is  its  volume  ? 

Ans.   ^,  or  25|  cu.  ft. 

7.  If  the  height  of  a  circular  cone  is  10  ft.,  what  must  be  the 
radius  of  its  base  in  order  that  the  volume  may  be  30  cu.  ft.  ? 

8.  A  frustum  of  a  cone  is  1  ft.  high   and  the  radii  of  its 
bases  are  respectively  9  ft.  and  4   ft. 
Find  its  volume. 

9.  If  r  and  B  are  the  radii  of  the 
bases  of  the  frustum  of  a  cone  and  I  is 
its  slant  height,  find  the  formula  for  its 
volume. 


274 


POLYHEDRONS 


[VII,  §  326 


PART  IV.   GENERAL  THEOREMS  ON  POLYHEDRONS 
SIMILARITY       REGULAR   SOLIDS       VOLUMES 

326.  Theorem  XVIII.  Two  triangular  pyramids  that  have 
a  trihedral  angle  of  the  one  equal  to  a  trihedral  angle  of  the  other 
are  to  each  other  as  the  products  of  the  edges  including  the  equal 
trihedral  angles.  h 


Given  the  triangular  pyramids  0-FGH  2indi  O'-F'G'II',  with 
the  trihedral  Z  0  =  trihedral  Z  0',  and  with  volumes  denoted 
by  V  and  V,  respectively. 

To  prove  that  V/V  =10F  •  OG  •  OH^/IO'F'  •  O'G'  -  O'lp-]: 

Proof.  Place  pyramid  O'-FG'H'  so  that  trihedral  Z  0'  will 
coincide  with  trihedral  Z.  0. 

Ytotr  H  and  H'  draw  HK  Sixid  H'K'  perpendicular  to  the 
plane  OFG. 

Then     V/r  =  [A  OFG  -  HKy[A  OF'G'  -  H'K'] 

=  [A  OFGy[A  OF'G'^  -  [HK/H'K^^.     Why  ? 
A  OFG        OF'OG 


But 


§  193 


A  0F'&      OF'  .  OG' 
Again,  let  the  plane  determined  by  HK  and  H'K'  intersect 
plane  OFG  in  line  OK'K. 

Then  rt.  A  OKH  ^  rt.  A  OK'H'.  W^hy  ? 

Therefore       HK/H'K'  =  OH/OW  Why  ? 

V  ^   OF'  OG       OH  ^     OF'OG -OH 

V  OF'  'OG'  '  OH'      O'F' '  O'G'  •  O'H'' 


Therefore 


VII,  §  330]  GENERAL  THEOREMS  275 

327.  Corollary  1.  Two  triangular  prisms  that  have  a  tri- 
hedral angle  of  the  one  equal  to  a  trihedral  angle  of  the  other  are 
to  each  other  as  the  products  of  the  edges  including  the  trihedral 
angles. 

[Hint.  Break  the  prism  up  into  triangular  pyramids,  and  use  §  326 
and  Theorem  H,  §  144.] 

328.  Corollary  2.  Two  parallelepipeds  that  have  a  trihedral 
angle  of  the  one  equal  to  a  trihedral  angle  of  the  other  are  to  each 
other  as  the  products  of  the  Qdges  including  the  trihedral  angle. 

329.  Similar  Tetrahedrons.  Two  tetrahedrons  (that  iS; 
triangular  pyramids)  are  said  to  be  similar  if  their  faces  are 
similar  each  to  each  and  similarly  placed. 

330.  Theorem  XIX.  The  volumes  of  two  similar  tetrahe- 
drons are  to  each  other  as  the  cubes  of  any  two  corresponding 
edges.  ^H 


Fig.  225 

Given  the  similar  tetrahedrons  0-i^G^^and  O'-F'G'H'  with 
the  volumes  denoted  by  V  and  V',  and  with  OF  and  OF'  two 
corresponding  edges. 

To  prove  that  V/  V  =  0F'/WF'\  §  §  158,  271 

Proof.  Trihedral  Z  0  =  trihedral  Z  0^  §  329 

Therefore 

V/r=    OF  'OG  'OH    ^  OF      OO      OH        .  3^5 
^  O'F'  .  O'G'  .  O'H'      O'F'  '  O'G'  *  O'H'' 

But  OF/O'F'  =  OG/O'G'  =  OH/0'H\  Why  ? 

Therefore 
V/r  =  {OF/ O'F')  {OF/ O'F')  {OF/ O'F')  =  0F'/WF'\ 


276 


POLYHEDRONS 


[VII,  §  331 


331.  Similar  Polyhedrons.     In  general,  similar  polyhedrons 

are  polyhedrons  which  have  the  same  number  of  faces  similar 
each  to  each  and  similarly  placed,  and  their  corresponding 
polyhedral  angles  equal. 

In  the  case  of  similar  tetrahedrons,  the  trihedral  angles  of 
the  one  are  necessarily  equal  to  those  of  the  other,  if  we  know 
only  that  the  faces  are  similar  each  to  each,  since  the  simi- 
larity of  the  faces  makes  the  three  face  angles  at  each  vertex 
equal  in  the  two  tetrahedrons,  by  §  158. 

By  §  330  and  Theorem  H,  §  144,  w^e  can  show  that  any  two 
similar  polyhedrons  are  to  each  other  as  the  cubes  of  any  two 
corresponding  edges. 

332.  The  Regular  Solids.  A  regular  polyhedron  is  one  whose 
faces  are  all  congruent  regular  polygons  and  whose  polyhedral 
angles  are  all  likewise  congruent.  Five  types  of  such  poly- 
hedrons are  represented  below. 


Fig,  226. —The  Five  Regular  Solibs 


We  shall  now  show  that  the  above  types  are  the  only  pos- 
sible types  of  regular  polyhedrons. 


VII,  §  333]  GENERAL  THEOREMS  277 

333.  Theorem  XX.  There  exist  only  Jive  different  types  of 
regular  polyhedrons. 

Proof.  The  proof  is  based  upon  two  facts :  (1)  that  any- 
polyhedral  angle  has  at  least  three  faces  and  (2)  that  the  sum 
of  the  face  angles  of  any  convex  polyhedral  angle  must  be 
less  than  360°.     (See  §  273.) 

Suppose  first  that  each  face  is  to  be  a  triangle.  Then,  from 
the  definition  of  a  regular  polyhedron,  the  triangle  must  be 
equilateral.  Each  of  its  angles  will  therefore  be  60°.  Con- 
sequently, by  statement  (2)  above,  polyhedral  angles  may  be 
formed  by  combining  three,  four,  or  five  such  angles,  but  no 
more  than  five  can  be  thus  used,  since  six  such  angles  amount 
to  360°,  while  seven  or  more  of  them  exceed  360°. 

Therefore,  not  more  than  three  regular  polyhedrons  are 
possible  having  triangles  as  faces.  The  three  that  are  possible 
are  the  regular  tetrahedron,  regular  octahedron,  and  regular 
icosahedron.    (See  Fig.  226.) 

Suppose  secondly  that  each  face  is  to  be  a  square.  Each 
face  angle  must  then  be  90°  and,  the  sum  of  four  such  angles 
being  360°,  it  follows  that  but  one  regular  polyhedron  is  pos- 
sible having  squares  as  sides.  The  cube  is  the  one  that  is 
possible. 

Thirdly,  suppose  that  each  face  is  to  be  a  regular  pentagon. 
Since  each  of  the  angles  of  such  a  figure  is  108°,  it  follows 
that  no  more  than  one  regular  polyhedron  is  possible  whose 
faces  are  pentagons.  The  one  that  is  possible  is  the  dodeca- 
hedron. 

We  can  proceed  no  farther,  for  the  sum  of  three  angles  of  a 
regular  hexagon  is  360°,  while  the  sum  of  three  angles  of  any 
regular  polygon  of  more  than  six  sides  is  greater  than  360°. 

Hence  the  theorem  is  proved. 

Note.  These  regular  solids  occur  in  nature  in  the  forms  of  a  variety 
of  crystals ;  but  not  all  crystals  are  regular  solids. 


278 


POLYHEDRONS 


[VII,  §  334 


334.  Models.  Models  of  the  five  possible  regular  polyhe- 
drons can  be  easily  constructed  as  follows : 

Draw  diagrams  on  cardboard  as  indicated  in  the  figures  below. 
Cut  these  out  and  then  cut  half  way  through  the  dotted  lines 
so  as  to  make  it  easy  to  fold  along  these  lines. 


Fig.  227 


Fold  on  the  dotted  lines  so  as  to  bring  the  edges  together,  sub- 
sequently pasting  strips  of  paper  over  the  edges  to  hold  the 
solid  in  position.  Models  of  the  tetrahedron,  octahedron,  and 
icosahedron  may  also  be  made  very  quickly  by  hinging  to- 
gether short  umbrella  wires  by  means  of  strong  copper  wires 
strung  through  the  end  holes,  joining  together  at  each  corner 
the  proper  number  of  rods.  The  student  may  show  that  each 
of  these  models  will  be  quite  rigid  when  completed. 


VII,  §  335] 


GENERAL  THEOREMS 


279 


335.  Theorem  XXI.  Cavalieri^s  Theorem.  If  two  solids 
are  included  between  the  same  pair  of  parallel  planes,  and  if 
every  section  of  one  of  the  solids  by  any  plane  parallel  to  one  of 
these  parallel  planes  is  equal  in  area  to  the  section  of  the  other 
solid  by  the  sanie  plane,  the  volumes  of  the  two  solids  are  equal. 


Fig.  228 

Outline  of  Proof.  The  two  solids  may  be  divided  into  a  large 
number  of  thin  slices  by  sections  parallel  to  the  two  including 
planes.  These  slices  may  be  thought  of  as  approximately 
cylindrical,  and  the  sum  of  all  the  slices  in  either  case  is  the 
volume  of  the  solid. 

The  bases  of  two  corresponding  slices  of  the  two  solids 
between  the  same  two  planes  are  equal  in  area,  by  hypothesis. 
It  is  therefore  apparent  that  the  volumes  of  the  two  correspond- 
ing slices  differ  as  little  as  we  please  if  their  thickness  is  suffi- 
ciently small.  It  can  be  shown  in  a  precise  manner  that  the 
total  volume  V  of  one  of  the  solids,  which  is  the  sum  of  all 
such  slices,  differs  from  the  total  volume  V  of  the  other  solid 
by  as  little  as  we  please. 

Hence,  as  in  §§  303,  311,  it  follows  that  V  =  V'. 


Note.     Observe  also  that  §§ 
what  precedes. 


303  are  essentially  special  cases  of 


280  POLYHEDRONS  [VII,  §  336 

336.   Theorem   XXII.    The   Prismoid   Formula.     If  any 

solid  S  of  a7iy  of  the  kinds  considered  in  this  Chapter  is  bounded 
by  two  parallel  plane  sections  B  and  T,  and  if  M  denotes  the  area 
of  another  section  parallel  to  and  midway  between  B  and  T,  the 
volume  V  of  S  is  given  by  the  formula : 

V=(B-\-  T  +  4.M)'h/(y, 
where  B,  T,  and  M  denote  the  areas  of  the  sections,  and  h  denotes 
the  distance  beticeen  B  and  T. 

The  proof  of  the  preceding  formula  consists  in  showing  that 
it  reduces  in  every  case  to  the  very  formula  for  volume  that 
has  already  been  proved  in  the  articles  above. 

Outline  of  Proof  for  Prisms  and  Cylinders.  In  these  cases,  all 
parallel  sections  are  equal  (§§  276,  310).  Hence  B=  T=  M; 
and  the  formula  to  be  proved  becomes  V=B'h,  which  we 
have  alread}*  proved  to  be  correct  (§§  296,  314). 

Outline  of  Proof  for  Pyramids  and  Cones.  In  these  cases  we 
know  that  the  area  of  any  section  parallel  to  the  base  B  is  pro- 
portional to  the  square  of  its  distance  from  the  vertex  (§§  301, 
322).     Hence,  since  J^f  is  at  a  distance  h/2  from  the  vertex, 

M      (h/2y      1  T.      A   nr 

—  =  ^  '    '  =-,  or  B  =  4:M. 
B         W         4' 

The  top  section  T  is  zero,  since  the  top  bounding  plane  meets 

a  pyramid  or  a  cone  in  just  one  point  on  the  vertex. 

Hence  the  formula  to  be  proved  becomes,  in  this  case, 

V=[_B+  r+  4  Jf]  .  V6  =  (J5  4-  0  4-  B)  -  h/6  =  Bh/3, 

which  we  know  to  be  correct.  §§  306,  323 

Outline  of  Proof  for  Frustums.  Given  a  frustum  of  a  pyramid  or  of 
a  cone,  let  H  be  the  distance  from  the  vertex  0  to  the  larger  of  the  two 
bounding  sections.  Let  B  represent  this  larger  section.  Then  H  —  h  is 
the  distance  from  O  to  the  other  bounding  section  T,  and  H  —  h/'2.  is  the 
distance  from  0  to  the  middle  section  M. 

We  know  that  the  volume  V  of  the  frustum  is 

V  =  \_BH -  T{H -  h)y^.  §§307,324 

Or,  since  T/B  =  {H-  hY/m,  §§  301, 322 

we  know  that 


VII,  §  337] 


GENERAL  THEOREMS 


281 


V  =  B[H  -{H-  hY/m^/Z  =  -^[3  H^-  SHh-\-  h'^:\h/3. 

Since  T/B  ={H-  hy^H^,  and  M/B  ={H-  h/2f/H^, 
the  formula  to  he  proved  may  be  written, 

H"  3 

This  is  equivalent  to  the  formula  that  we  know  to  be  correct ;  hence  the 
theorem  is  proved. 

337.  Uses  of  the  Prismoid  Formula.  The  prismoid  formula  is  a  con- 
venient means  of  remembering  the  volumes  of  a  variety  of  solids.  We 
shall  see  in  Chapter  VIII  that  it  holds  for  spheres  and  frustums  of  spheres 
as  well  as  for  the  solids  of  this  chapter. 

It  also  holds  for  any  solid  bounded  by  two  parallel  planes,  made  up  by 
joining  together  pyramids,  prisms,  etc. ,  bounded  by  the  same  two  planes  ; 
such  a  solid  is  called  a  prismoid. 


Fig.  229 

The  formula  is  used  very  extensively  by  engineers  to  estimate  the 
volumes  of  various  objects,  such  as  the  volume  of  a  hill,  or  the  volume 
of  a  metal  casting.  Since  the  same  formula  holds  for  such  a  large  variety 
of  solids,  it  is  reasonably  safe  to  use  it  without  even  stopping  to  see 
which  of  these  solids  really  resembles  the  object  whose  volume  is  desired. 

It  is  shown  in  more  advanced  books  that  the  same  formula  holds  when- 
ever the  area  of  the  section  by  any  plane  parallel  to  the  bounding  planes 
is  proportional  to  the  square  of  the  distance  from  some  fixed  point  to  that 
plane.  Many  solids  not  mentioned  otherwise  in  elementary  geometry 
satisfy  this  requirement. 


282  POLYHEDRONS  [VII,  §  337 

MISCELLANEOUS   EXERCISES.     CHAPTER  VII 

1.  Show  that  every  section  of  a  cylinder  made  by  a  plane 
passing  through  an  element  is  a  parallelogram.  What  is  the 
section  when  the  cylinder  is  a  right  cylinder  ? 

2.  Show  that  every  section  of  a  cone  made  by  a  plane 
through  the  vertex  is  a  triangle.  What  is  the  section  when 
the  cone  is  a  right  cone  ? 

3.  If  a,  h,  and  c  are  the  dimensions  of  a  parallelepiped, 
show  that  the  length  of  its  diagonal  is  Va^  +  &^  +  c^ 

4.  How  long  an  umbrella  will  go  into  a  trunk  measuring 
32  in.  by  17  in.  by  21  in.,  inside  measure,  (a)  if  the  umbrella 
is  laid  on  the  bottom  ?  (6)  if  it  is  placed  diagonally  between 
opposite  corners  of  the  top  and  bottom  ? 

5.  Find  the  volume  of  a  pyramid  whose  base  is  a  rhombus 
6  in.  on  a  side  and  whose  height  is  6  in.,  if  one  angle  of  the 
rhombus  is  60°. 

6.  The  Great  Pyramid  in  Egypt  is  about  480  ft.  high  and 
its  base  is  a  square  measuring  about  764  ft.  on  a  side.  Find 
approximately  its  volume  in  cubic  yards. 

7.  Water  is  poured  into  a  cylindrical  reservoir  25  ft.  in 
diameter  at  the  rate  of  300  gallons  a  minute.  Find  the  rate 
(number  of  inches  per  minute)  at  which  the  water  rises  in  the 
reservoir  (1  gal.  =  231  cu.  in.). 

8.  A  copper  teapot  is  9|-  in.  in  diameter  at  the  bottom, 
8  in.  at  the  top,  and  11  in.  deep.  Allowing  42  sq.  in.  for  locks 
and  waste,  how  much  metal  is  required  for  its  construction, 
excluding  the  cover  ? 

9.  A  conical  spire  has  a  slant  height  of  60  ft.  and  the 
perimeter  of  the  base  is  50  ft.     Find  the  lateral  surface. 

10.  How  many  cubic  inches  of  lead  are  there  in  a  piece  of 
lead  pipe  2  yd.  long,  the  outer  diameter  being  2  in.  and  the 
thickness  of  the  lead  being  \  of  an  inch  ? 


VII,  §337]        MISCELLANEOUS  EXERCISES  283 

11.  The  chimney  of  a  factory  has  approximately  the  shape 
of  a  frustum  of  a  regular  pyramid.  Its  height  is  75  ft.  and 
its  upper  and  lower  bases  are  squares  whose  sides  are  5  ft. 
and  8  ft.  respectively.  The  flue*is  throughout  a  square  whose 
side  is  3  ft.  How  many  cubic  feet  of  material  does  the 
chimney  contain  ?  Assuming  that  a  brick  is  8  in.  long,  3J  in. 
wide,  and  2J  in.  thick  (as  is  ordinarily  the  case),  estimate  the 
number  of  bricks  in  such  a  chimney. 

12.  Compare  the  lateral  areas,  the  total  areas,  and  the 
volumes  of  (1)  a  right  circular  cylinder  and  a  right  circular 
cone  having  equal  bases  and  altitudes,  (2)  a  regular  pyramid 
and  a  regular  prism  having  congruent  bases  and  equal  altitudes. 

13.  A  standard  rain-gauge  is  made  by  inclosing  a  tube  B 
in  the  interior  of  a  can  ACDE  and  connecting  the  mouth  of 
the  tube  to  the  mouth  of  the  can  by  a  fun- 
nel FOHI.  The  amount  of  water,  measured 
in  inches  (depth),  that  has  fallen  in  the 
vicinity  of  the  gauge  is  determined  by  read- 
ing the  height  of  the  water  in  the  tube  B. 
Find  a  formula  for  the  amount  of  rain  that 
has  fallen  in  terms  of  the  height  h  of  the 
water  in  the  tube  B,  the  radius  r  of  the  tube, 
and  the  radius  R  of  the  can.     Ans.  hr^/R^. 

14.  If  one  of  the  edges  of  a  tetrahedron  is  1  in.  long,  how 
long  will  be  the  corresponding  edge  of  a  similar  tetrahedron 
of  8  times  the  volume?  Answer  the  same  question  for  the 
case  in  which  the  new  tetrahedron  is  to  have  half  the  volume 
of  the  original.  Ans.  2  in. ;  l/-v^2  =  .79  in. 

15.  It  is  usual  to  state  the  diameter  d  of  a  tube  in  inches, 
and  the  area  A  of  its  surface  in  square  feet.  Show  that  the 
formula  used  by  engineers  : 

^  =  0.2618  dZ 
gives  very  nearly  the  correct  value  in  square  feet,  if  d  is  meas- 
ured in  inches,  and  the  length  I  is  measured  in  feet. 


CHAPTER   VIII 

THE   SPHERE 

PART   I.     GENERAL   PROPERTIES 

338.  Spheres.  A  sphere  is  a  portion  of  space  bounded  by  a 
surface  such  that  all  straight  lines  to  it  from  a  fixed  point 
within  are  equal. 

The  fixed  point  within  the  sphere  is  called  its  center;  a 
line  segment  joining  the  center  to  any  point  on  the  surface  is 


Fig.  230 

a  radius;   a  line  segment  drawn  through  the  center  and  ter- 
minated at  both  ends  by  the  surface  is  a  diameter. 
It  follows  from  these  definitions  that : 

(a)  The  radii  of  a  sphere,  or  equal  spheres,  are  equal. 

(b)  The  diameters  of  a  sphere,  or  of  equal  spheres,  are  equal. 

(c)  Spheres  having  equal  radii,  or  equal  diameters,  are  equal. 

(d)  A  sphere  may  he  generated  by  the  revolution  of  a  semicircle 
about  its  diameter. 

284 


VIII,  §  339J 


GENERAL  PROPERTIES 


285 


EXERCISES 

1.  What  is  the  locus  of  the  points  that  are  2  in.  from  the 
surface  of  a  sphere  whose  radius  is  4  in.  ? 

2.  Show  that  the  distance  from  the  center  of  a  sphere  to 
a  point  outside  the  sphere  is  greater  than  the  radius.  (Use 
Ax.  10.)     State  the  converse.     Is  it  true  ? 

3.  If  two  spheres  have  the  same  center,  they  are  called 
concentric.  Show  that  one  of  two  concentric  spheres  lies 
wholly  within  the  other. 

4.  Show  that  if  the  center  of  each  one  of  two  given  spheres 
lies  on  the  surface  of  the  other,  their  radii  are  equal. 

5.  Show  by  §  77,  that  a  plane  perpendicular  to  a  diameter 
of  a  sphere  at  its  extremity  has  only  one  point  in  common 
with  the  sphere. 


339.  Tangent  Planes  and  Lines.  A  plane  that  has  only 
one  point  in  common  with  a  sphere  is  called  a  tangent  plane  to 
the  sphere.  A  line  that  has  only  one  point  in  common  with  a 
sphere  is  called  a  tangent  line  to  the  sphere.  In  either  case, 
the  single  common  point  is  called  the  point  of  tangency. 


286 


THE  SPHERE 


[VIII,  §  340 


Fig.  231 


340.  Theorem  I.  A  plane  perpendicular  to  a  diameter  of  a 
sphere  at  one  of  its  extremities  is  tangent  to  the  sphere. 

Outline  of  Proof.  Let  MN  be  a 
plane  perpendicular  to  a  diameter  PP' 
at  P.  Connect  any  point  A  of  MN  to 
the  center  of  the  sphere  0.  Show,  by 
§  77,  that  0A>  OP;  whence,  by  § 338, 
A  cannot  be  on  the  sphere,  so  that  P 
is  the  only  point  of  the  plane  on  the 
sphere. 

341.  Corollary  1.  (Converse 
of  Theorem  I.)  If  a  plane  is 
tangent  to  a  sphere,  it  is  perpen- 
dicular to  the  radius  draivn  to  the 
point  of  contact. 

[Hint.  Show,  by  Ax.  10,  that  the  radius  is  shorter  than  any  other  line 
drawn  from  the  center  to  the  plane.     Then  use  §  77.] 

342.  Corollary  2.  A  straight  line  perpendicular  to  a  diameter 
of  a  sphere  at  one  of  its  extremities  is  tayigent  to  the  sphere;  and 
conversely.      [Hint.     Use  §§  254,  340  for  direct,  and  §  116  for  converse.] 

343.  Corollary  3.  All  of  the  straight  lines  tangent  to  a  sphere 
at  a  given  point  lie  in  the  plane  tangent  to  the  sphere  at  that 
point,     [Hint.     Use  §  264.] 

EXERCISES 

1.  What  is  the  locus  of  a  point  in  space  at  a  given  distance 
from  a  given  point  ? 

2.  Prove  that  if  two  lines  are  tangent  to  a  sphere  at  the 
same  point;  their  plane  is  tangent  to  the  sphere. 

[Hint.     Connect  this  with  one  of  the  corollaries  on  this  page.] 

3.  All  lines  tangent  to  a  sphere  from  the  same  point  are 
equal. 

[Hint.  Connect  the  center  of  the  sphere  with  the  given  point  and  with 
two  or  more  points  of  tangency.] 


VIII,  §  344] 


GENERAL  PROPERTIES 


287 


344.  Theorem  II.    Every  section  of  a  sphere  made  by 
a  plane  is  a  circle. 


7\: 


A 


Fig.  232 

Given  the  sphere  whose  center  is  0,  cut  by  a  plane  in  the 
section  AMBN. 

To  prove  that  section  AMBN  is  a  circle. 

Proof.  Draw  OQ  J- section  AMBN\  join  Q  to  C  and  D, 
any  two  points  in  the  perimeter  of  the  section ;  draw  OC  and 
OD. 

In  the  rt.  A  OQC  and  OQD, 

OQ  =  OQ,  and    OC  =  OD.  Why  ? 

Therefore        ,  A  OQO  ^  A  OQD.  Why  ? 

Therefore  QC  =  QD.  Why? 

Since  (7  and  D  are  a^ii/  two  points  on  the  perimeter  of  the 
section,  all  points  on  the  perimeter  of  the  section  are  equally 
distant  from  Q.      Therefore  section  AMBC  is  a  circle.      §  103 

EXERCISES 

1.  If,  in  Fig.  232,  the  radius  of  the  sphere,  OC,  is  10  in., 
and  the  distance  OQ  from  the  center  0  to  the  plane  AB  is 
6  in.,  find  the  radius  CQ  of  the  circle. 

2.  If,  in  Fig.  232,  the  distances  CQ  and  OC  are  given,  show 
how  to  find  the  distance  OQ. 


288 


THE  SPHERE 


[Vni,  §  345 


345.  Great  and  Small  Circles.  A  circle  on  the  sphere  whose 
plane  passes  through  the  center  is  called  a  great  circle  of  the 
sphere,  as  CD,  Fig.  233. 


Fig.  233 


A  circle  on  the  sphere  whose  plane  does  not  pass  through  the 
center  is  called  a  small  circle  of  the  sphere,  as  AB,  Fig.  233. 

The  axis  of  a  circle  of  a  sphere  is  the  diameter  of  the  sphere 
which  is  perpendicular  to  the  plane  of  the  circle. 

The  poles  of  a  circle  of  a  sphere  are  the  extremities  of  the 
axis  of  the  circle. 

346.  Corollary  1.  Through  any  three  points  on  the  surface 
of  a  sphere  one  and  only  one  circle  of  the  spheres  may  he  drawn. 

[Hint.     Use  4,  §  241.] 

347.  Corollary  2.  Through  any  two  points  on  the  surface  of 
a  sphere  a  great  circle  may  he  drawn.  It  follows  from  4,  §  241, 
that  there  is  one  and  only  one  such  great  circle  through  the  two 
given  points,  unless  they  lie  at  the  opposite  ends  of  a  diameter. 

348.  Distance  on  a  Sphere.  By  the  distance  between  two 
points  on  the  surface  of  a  sphere  is  meant  the  length  of  the 
shorter  arc  of  the  great  circle  joining  them.  It  can  be  shown 
that  this  is  the  shortest  path  on  the  surface  of  the  sphere  be- 
tween the  two  points. 


VIII,  §  348] 


GENERAL  PROPERTIES 


289 


EXERCISES 

1.  If  we  consider  the  earth  as  a 
sphere,  what  kind  of  circles  are 
the  parallels  of  latitude?  the 
equator  ?  the  meridians  ? 

2.  Prove  that  the  axis  of  a  small 
circle  of  a  sphere  passes  through 
the  center  of  the  circle ;  and  con- 
versely, a  diameter  of  the  sphere 
through  the  center  of  a  small  circle 
is  the  axis  of  that  small  circle. 

3.  Prove  that  in  the  same  sphere,  or  in  equal  spheres,  all 
great  circles  are  equal. 

4.  The  radius  of  a  sphere  is  10  in.  Find  the  area  of  a 
section  made  by  a  plane  5  in.  from  the  center. 

5.  The  area  of  a  section  of  a  sphere  7  in.  from  the  center 
is  288  TT  sq.  in.     Find  the  area  of  a  section  4  in.  from  the  center. 

6.  Prove  that  in  the  same  sphere,  or  in  equal  spheres,  if  two 
sections  are  equal,  they  are  equally  distant  from  the  center,  and 
conversely. 


1 

•■'    9.     Q 

>  \ 

& 


(3  t 


7.   Prove  that  any  two  great  circles  of  a  sphere  bisect  each 
other. 


290 


THE  SPHERE 


[VIII,  §  349 


349.    Theorem  III.     All  points  in  the  circumference  of  a  circle 
of  a  sphere  are  equally  distant  from  either  one  of  its  poles. 


- 

A,l 

:•: .  1 

:D 

/  \ 

-4 

' 

I 

o 

1 

-- 

j 

j 

Fig.  234  (a) 


Fig.  234  (Jb) 


Given  any  two  points  A  and  B  in  the  circumference  of  the 
circle  ABC,  and  P  and  P',  the  poles  of  ABC. 

To  prove  that  PA  =  PB,  and  P^4  =  jFb. 

Proof.     Draw  the  great  circles  PAP'  and  PBP'. 

Let  D  be  the  intersection  of  the  axis  PP'  with  the  plane 
ABC.     Draw  the  straight  lines  AD,  BD,  PA  and  PB. 

Now  PD  =  PD,  and  DA  =  DB,  Why  ? 

and  Z  PDA  =  Z  PDB  =  90°.  Why  ? 

Hence  chord  PA  =  chord  PB,  Why  ? 

Therefore  PA  =  PB.  Why  ? 

In  the  same  way  it  may  be  proved  that  P'A  =  P'B. 

Note.  The  manner  in  which  circles  may  be  drawn  on  a  sphere  is 
illustrated  by  Fig.  234  (6).  If  one  end  of  a  string  is  held  at  any  point  on 
the  sphere,  while  a  pencil  attached  to  the  other  end  is  moved  around  the 
sphere,  keeping  the  string  taut,  the  end  of  the  pencil  describes  a  circle  on 
the  sphere,  by  Theorem  III. 

The  various  figures  drawn  in  this  chapter  can  be  reproduced  on  the 
surface  of  an  actual  sphere,  by  this  method  of  drawing  the  circles. 


VIII,  §  351] 


GENERAL  PROPERTIES 


291 


350.  Polar  Distance.  The  polar  distance  of  a  circle  of  a 
sphere  is  the  distance  on  the  sphere  (§  348)  from  its  nearest 
pole  to  any  point  of  the  circumference,  as  PA  or  PB  in  Fig.  234. 

A  quadrant  is  one  fourth  part  of  the  circumference  of  a 
great  circle ;  i.e.  an  arc  of  90°  on  a  great  circle. 


351.   Corollary  1. 

quadrant. 


The  polar  distance  of  a  great  circle  is  a 


Fig.  235 


[Hint.  Let  ABC  be  a  great  circle.  Then  its  center  0  is  also  the 
center  of  the  great  circle  PBP'.  Hence  the  arc  PB  measures  the  right 
angle  POB.'\ 


EXERCISES 

1.  What  is  the  locus  of  all  the  points  on  the  surface  of  the 
earth  at  a  quadrant's  distance  from  the  north  pole  ?  from  the 
south  pole  ?  from  the  equator  ?    See  the  figure  for  Ex.  1,  p.  289. 

2.  The  distance  of  the  plane  of  a  certain  small  circle  from  the 
center  of  a  sphere  is  one  half  the  radius  of  the  sphere.  If  the 
diameter  of  the  sphere  is  12  in.,  find  the  polar  distance  of  the 
small  circle  in  degrees  and  in  inches.  Ans.  60°  ;  2  tt  in. 

3.  Show  that  a  great  circle  on  the  earth  whose  poles  lie  on 
the  equator  passes  through  the  north  pole. 


292 


THE  SPHERE 


[VIII,  §  352 


352.    Problem  I.     To  determine  the  radius  of  a  given  material 
sphere. 


"' 

^-^             ! 

e 

^•"^IfeL 

.0 

0 

/ 

\ 

1 

Fig.  236  (a) 


Fig.  236  (&) 


Given  any  material  sphere,  OPQ. 

To  find  its  radius. 

Construction.  Take  any  point  P  on  the  surface  of  the  sphere 
as  a  pole,  and  describe  the  circle  ABC. 

Take  any  three  points  on  this  circle,  as  A,  B,  C. 

By  means  of  the  compasses  construct  on  paper  or  on  the 
blackboard  the  triangle  A'B'C  congruent  to  the  triangle 
ABC. 

Circumscribe  a  circle  around  AA'B'C,  and  let  D'  be  the 
center  of  this  circle. 

Draw  D''A''  equal  to  the  radius  lyA'. 

Through  D"  draw  an  indefinite  line  P'Q'  perpendicular  to 

ly'A". 

Erom  A"  lay  off  with  compasses  A'^P'  equal  to  line  AP. 

At  A"  erect  a  perpendicular  to  A"P'  and  extend  it  to  meet 
P'Q'  at  Q\ 

Then  P'^  is  the  diameter  and  P'Q'/2  is  the  radius  of  the 
given  sphere. 

[The  proof  is  left  to  the  student.  ] 


VIII,  §  353] 


GENERAL  PROPERTIES 


293 


353.    Problem  II.     To  construct  a  sphere  through  four  given 
points  not  all  in  the  same  playie. 


Fig.  237 

Given  the  four  points  A,  B,  C,  D  not  all  in  the  same  plane. 
To  construct  a  sphere  that  passes  through  A,  B,  C,  and  D. 

Construction.  At  E,  the  middle  point  of  ABy  erect  a  plane 
QEP  perpendicular  to  AB.  Likewise,  let  PFR  be  a  plane 
perpendicular  to  .BO  at  its  middle  point  F\  and  let  QOR  be  a 
plane  perpendicular  to  -BZ>  at  its  middle  point  G. 

Let  0  be  the  point  common  to  all  three  planes  QEPj  PFR, 
and  QOR. 

With  0  as  center,  and  OA  as  radius,  draw  a  sphere. 

This  is  the  required  sphere  passing  through  A,  B,  C,  and  D. 

Proof.  The  plane  QEP  is  the  locus  of  all  points  equidistant 
from  A  and  B.  Why  ? 

Likewise,  PFR  is  the  locus  of  points  equidistant  from  B 
and  C;  and  QGR  is  the  locus  of  points  equidistant  from  B 
and  D. 

The  planes  QEP  and  PFR  meet  in  a  line  OP.  Why  ? 

The  line  OP  meets  the  plane  QOR  in  a  single  point  0. 

Why? 

Therefore  0  is  equidistant  from  A,  B,  C,  D.  Why  ? 

Moreover,  0  is  the  only  point  equidistant  from  A,  B,  O,  D. 

Why? 


294 


THE  SPHERE 


[Vin,  §  354 


354.  Inscribed  and  Circumscribed  Spheres.  A  sphere  is 
said  to  be  circumscribed  about  any  polyhedron  when  the  ver- 
tices of  the  polyhedron  all  lie  on  its  surface. 


Fig.  238.     Circumscribed  Sphere 


A  sphere  is  said  to  be  inscribed  in  any  polyhedron  when  it 
is  tangent  to  each  of  the  faces  of  the  polyhedron. 


/ 

aD 

/ 

\ 

A 

''^i 

"^ 

Fia.  239.      Inscribed  Sphere 

355.    Corollary  1.     One  and  only  one  sphere  may  he  circum- 
scribed about  any  given  tetrahedron  (triangular pyramid). 

[Hint.    Pass  a  sphere  through  the  four  vertices  as  in  §  353.    Notice 
that  the  four  vertices  cannot  all  lie  in  one  plane.] 


VIII,  §356]  GENERAL  PROPERTIES  295 

356.  Problem  III.  To  inscribe  a  sphere  in  a  given  tetrahe^ 
dron  (triangular  pyramid). 

Given  the  tetrahedron  ABCD,  Fig.  239. 

To  construct  the  sphere  inscribed  in  it. 

Construction  and  Proof.  Bisect  the  dihedral  angles  whose 
edges  are  BC,  CD,  and  DB,  by  the  planes  BOC,  CED,  and 
DEB,  respectively. 

The  plane  BOC  is  the  locus  of  the  points  equidistant  from 
the  faces  BCD  and  BAC;  the  plane  CED  is  the  locus  of  the 
points  equidistant  from  the  faces  BCD  and  CAD ;  and  the  plane 
DEB  is  the  locus  of  the  points  equidistant  from  the  faces  BCD 
and  DAB.  Why  ? 

The  intersection  0  of  these  planes  is  equidistant  from  the 
four  faces  of  the  tetrahedron.  Hence  the  sphere  whose  center 
is  0  and  whose  radius  is  the  perpendicular  distance  OE  from 
O  to  the  face  ABC,  is  tangent  to  each  of  the  faces;  it  is 
therefore  inscribed  in  the  tetrahedron. 

No  other  sphere  exists  that  is  inscribed  in  the  tetrahedron, 
for  no  other  point  than  0  is  equidistant  from  the  four  faces. 

Why? 
EXERCISES 

1.  By  means  of  an  instrument  called  a  spherometer,  the 
distances  AD  and  DP,  Fig.  236,  can  be  measured  directly. 
Show,  by  §  162,  how  to  find  the  radius  from  these  values. 

2.  Show  that  the  process  of  §  352  can  be  used  to  find  the 
radius  of  a  sphere,  if  only  a  piece  of  the  sphere  is  available,  as 
in  the  case  of  a  glass  lens. 

3.  Show  that  four  points  in  space  determine  a  sphere,  pro- 
vided they  do  not  lie  in  one  plane. 

4.  Show  that  a  sphere  is  determined  if  any  circle  that  lies 
on  it  and  one  pole  of  that  circle  are  given. 

5.  Show  that  any  two  circles  of  a  sphere  completely  de- 
termine the  sphere. 


296 


THE  SPHERE 


[VIII,  §  357 


PART  II.     SPHERICAL   ANGLES  — TRIANGLES — 
POLYGONS 

357.  Spherical  Angles.  The  line  tangent  to  a  great  circle 
of  a  sphere  at  any  point  is  a  tangent  to  the  sphere  at  that 
point ;  for  it  touches  the  sphere  in  only  one  point  (§  339). 

The  angle  formed  by  the  intersection  of  two  great  circles  is 
called  a  spherical  angle.  It  is  defined  to  be  equal  to  the  angle 
formed  by  the  tangents  to  the  two  great  circles,  at  their  point 
of  intersection,  as  the  angle  CPD,  Fig.  240. 

358.  Theorem  IV.  Tlie  angle  hetioeen  two  great  circles 
is  measured  hy  the  arc  of  a  great  circle  described  from 
its  vertex  as  a  pole  and  included  hetiveen  its  sides. 


Fig.  240 

Given  the  great  circles  PAP  and  PBP'  intersecting  at  P,  and 
AB  the  arc  of  a  great  circle  described  with  P  as  a  pole. 

To  prove  that  ^B  is  the  measure  of  Z  APE. 

Proof.     Draw  the  radii  OAj  OB,  and  the  tangents  PC,  PD. 
Then  OA  II  PC  and  OB  II  PD.  Why  ? 

Hence  Z.  AOB  =^  Z.  CPD.  Why  ? 

But  Z.  AOB  is  measured  by  the  arc  AB',  hence  Z  CPD  is 


VIII,  §  360]        TRIANGLES  AND  POLYGONS 


297 


measured  by  the  arc  AB.  It  follows  that  the  spherical  angle 
APB,  which  is  equal  to  Z  CPD  by  definition  (§  357),  is  meas- 
ured by  the  arc  AB. 

359.  Corollary  1.  The  spherical  angle  between  two  great  circles 
is  equal  to  the  plane  angle  of  the  dihedral  angle  formed  by  the 
planes  of  the  tivo  great  circles. 

360.  Spherical  Triangles  and 
Polygons.  A  spherical  polygon  is 
a  portion  of  a  spherical  surface 
bounded  by  three  or  more  arcs  of 
great  circles;  as  ABODE,  Fig.  241. 

The  bounding  arcs  of  great  cir- 
cles are  called  the  sides  of  the 
spherical  polygon ;  their  intersec- 
tions, the  vertices ;  and  the  angles 
formed  by  the  sides  at  the  ver- 
tices, the  angles  of  the  spherical 
polygon. 

A  diagonal  of  a  spherical  polygon  is  an  arc  of  a  great  circle 
joining  any  two  non-adjacent  ver- 
tices. 

A  spherical  triangle  is  a  spherical 
polygon  of  three  sides,  as  ABC, 
Fig.  242. 

The  words  isosceles,  equilateral, 
acute,  right,  and  obtuse  are  applied 
to  spherical  triangles  in  precisely 
the  same  way  as  to  plane  triangles. 

Thus,  in  Fig.  242,  the  spherical 
triangle  ABC  is  isosceles   if  the 
two    sides,    as   AB   and   BC,   are 
equal;  the  triangle  is  equilateral  if  AB  —  BC  =  AC\  the  tri- 
angle is  a  right  triangle  if  any  one  angle  is  a  right  angle ;  etc. 


Fig.  241 


Fig.  242 


298 


THE  SPHERE 


[VHI,  §  361 


361.  Relation  to  Central  Polyhedral  Angles.  The  planes 
of  the  arcs  of  the  great  circles  forming  the  sides  of  a  spher- 
ical polygon  meet  at  the  center  of  the  sphere  and  form  a  poly- 
hedral angle,  as  0-ABCD. 


Fig.  243 

This  polyhedral  angle  and  the  spherical  polygon  are  so 
closely  related  that  the  student  can  easily  prove  the  following 
statements : 

(a)  The  sides  of  a  spherical  polygon  have  the  same  measures  as 
the  corresponding  face  angles  of  the  polyhedral  ajigle. 

(b)  The  angles  of  the  spherical  polygon  have  the  same  measures 
as  the  corresponding  dihedral  angles  of  the  polyhedral  angle. 

Thus,  sides  AB,  BO,  etc.,  of  the  spherical  polygon  ABCD 
have  the  same  measures  as  face  A  AOB,  BOO,  etc.,  of  polyhe- 
dral Z  O-ABOD;  and  spherical  A  ABO,  BOD,  etc.,  have  the 
same  measures  as  the  dihedral  A  whose  edges  are  OB,  00,  etc. 

(c)  Any  angle  of  a  spherical  polygon  (or,  the  corresponding 
dihedral  angle  of  the  polyhedral  angle)  is  measured  by  the  arc  of 
a  great  circle  described  with  the  vertex  of  the  angle  as  pole  and 
terminated  by  the  sides.     See  §§  858,  359. 

In  general,  any  fact  proved  for  the  sides  and  the  angles  of  a 
spherical  polygon  is  true  also  for  the '  corresponding  face  angles 
and  dihedral  angles  of  the  corresponding  central  polyhedral  angle. 


VIII,  §  363]       TRIANGLES  AND  POLYGONS 


299 


362.   Theorem  V.     The  sum  of  any  two  sides  of  a  spherical 
triangle  is  greater  than  the  third  side.     [Compare  §  272.] 


/^: 

x^.. 

\ 

0                  H ; 

Fig.  244 

Given  the  spherical  A  ABC. 

To  prove  that  AB -\- BC  >  62. 

Proof.  Z  AOB  +  Z  BOO  >  Z  COA, 

Z  AOB  is  measured  by  AB, 
Z  BOG  is  measured  by  BO, 
Z  COA  is  measured  by  CA. 

Therefore      AB -{- BC  >  CA. 


363.   Theorem  VI.    The  sum  of 

the  sides  of  any  convex  spherical 
polygon  is  less  than  360°.  [Com- 
pare §  273.] 

Given     the     spherical    polygon 
ABCD. 

To  prove  that 

AB-{-BC+cb-\-DA<  360°. 
[Hint.     Make  use  of  §  273.] 


§272 


Why? 


Fig.  245 


300 


THE  SPHERE 


[Vni,  §  364 


EXERCISES 
1.    Show  that  any  side  of  a  spherical  polygon  is  less  than  180°. 


2.    In  the  spherical  A  ABC,  AB  =  35°,  and  BC 
tween  what  limits  must  CA  lie  ? 


75°.     Be- 


3.  Three  of  the  sides  of  a  spherical  quadrilateral  are  respec- 
tively 88°  IT,  70°  36',  and  50°  33'.  Between  what  limits  must 
the  fourth  side  lie  ? 


364.  Polar  Triangles.  If  from  the  vertices  of  a  spherical 
triangle  as  poles  arcs  of  great  circles  are  drawn,  these  arcs 
form  a  second  triangle  which  is  called  the  polar  triangle  of 
the  first. 


Fig.  24() 

Thus,  if  A,  B,  G,  the  vertices  of  the  spherical  A  ABC,  are 
the  poles  of  the  arcs  B'C,  A'C,  A'B',  forming  the  spherical 
A  A'B'C,  then  A'B'C  is  the  polar  triangle  of  ABC. 

If  the  entire  circles  be  drawn,  they  will  intersect  so  as  to 
form  eight  spherical  triangles,  but  the  polar  of  the  given 
triangle  ABC  is  that  one  of  the  eight  triangles  whose  vertices 
lie  on  the  same  side  of  the  arcs  of  the  given  triangle  as  the 
corresponding  vertices  of  the  given  triangle,  and  no  side  of 
which  is  greater  than  180°. 


VIII,  §365]       TRIANGLES  AND  POLYGONS 


301 


365.   Theorem  VII.     If  one  spherical  triangle  is  the  polar  of 
another  y  then  the  second  is  the  polar  of  the  first. 


Fig.  247 

Given  A  A'B'C,  the  polar  of  A  ABC. 

To  prove  that  A  ABC  is  the  polar  of  A  A'B'C. 

Proof.     A  is  the  pole  of  B'C,  and  (7  is  the  pole  of  A'B' ; 

Given. 

hence  B'  is  at  a  quadrant's  distance  from  A  and  (7,  so  that  B'  is 
the  pole  of  AG.  §  249 

Similarly,  A'  is  the  pole  of  BC,  and  C"  is  the  pole  of  AB. 

Therefore  ABC  is  the  polar  triangle  of  A'B'C\  §  364. 

EXERCISES 

1.  Show  that  if  one  side  of  a  spherical  triangle  on  the  earth's 
surface  is  on  the  equator,  one  vertex  of  the  polar  triangle  is  either 
at  the  north  pole  or  at  the  south  pole. 

2.  Show  that  if  one  vertex  of  a  triangle  on  the  earth  is  at 
the  north  pole,  one  side  of  the  polar  triangle  is  on  the  equator. 

3.  Show  that  if  one  vertex  of  a  triangle  on  the  earth  is  at 
the  north  pole,  and  if  one  side  of  the  triangle  is  on  the  equator, 
the  polar  triangle  also  has  one  vertex  at  the  north  pole  and 
one  side  along  the  equator. 


302 


THE  SPHERE 


[Vni,  §  366 


366.  Theorem  VIII.  In  tivo  polar  triangles,  each  angle  of 
the  one  is  measured  by  the  supplement  of  the  side  opposite  to  it  in 
the  other. 


^/€^~<^- 

/^c 

r--x^  '   V 

/   / 

a\       dV' 

h                 X    -B), 

1        b     r     .          5>\/'' 

^        V'^^-"""''''^     -'      ■ 

■  -7^             C^-'''                       ' 

.^'^'                               1 

Fig.  248 

Given  the  polar  triangles  ABC  and  AB'O,  with  the  sides 
denoted  by  a,  6,  c,  and  a',  h' ,  c',  respectively. 
To  prove  that 

(a)  Z  ^  +  a' =  180°,  Z5  +  6'  =  180°,  Z(7+c'  =  180°; 

lh)/.A  +  a  =180°,   ZJB'  +  6  =  180°,  ZC"  +  c  =  180°. 

Proof.     Let  AB  and  AC  (prolonged,  if  necessary)  intersect 
B'C  at  D  and  E,  respectively. 


Then 
Therefore 
That  is 


C^D 


Why? 
Why? 


(c)  §  361 


90°,  andjB;J5'  =  90°. 
O'i)  +  ^^'  =  180°. 
OE  ■\-ED-\-ED  +  DB'  =  180°, 
or,  ^Z>  +  a'  =  180°. 

But  ED  is  the  measure  of  Z  A. 
Therefore  Z^  +  a'  =  180°. 

In  a  similar  manner  Z  J5  +  &'  =  180°,  and  Z  (7+  c'  =  180°. 
The  proof  of  (6)  is  left  for  the  student. 

EXERCISE 

1.    If  the  angles  of  a  spherical  triangle  are  70°,  90°,  and  80°, 
respectively,  find  the  sides  of  the  polar  triangle  (in  degrees). 


VIII,  §  368]       TRIANGLES  AND  POLYGONS 


303 


367.    Theorem  IX.     The  sum  of  the  angles  of  a  spherical  tri- 
angle is  greater  tha,n  180°  and  less  than  540°. 


Fig.  249 

Given  the  spherical  A  ABC  with  the  sides  a,  b,  and  c.  - 
To  prove  that  ZA-\-ZB-\-ZO>  180°  and  <  540°. 
Proof.     Let  AA'B'G\  with  its  sides  denoted  by  a',  h',  and  c', 
be  the  polar  of  A  ABC. 

Then  ZA  +  a'  =  180°,  Z  5  +  6'  =  180°,  AC+c'  =  180°. 
Therefore  Z.  A -\- A  B -\- /.  C  +  a' +  ?>' +  c' =  540°.         Why? 
But  '    a'  4-  &'  +  c'  <  360°.  §  363 

Therefore  ZA^-ZB-\-ZC>  180°.  Why  ? 

Again  a'  +  5'  +  c'  >  0°. 

Therefore  Z  ^  +  Z5  + Z  C<  540°.  Why? 

368.  Corollary  1.  In  a  spherical  triangle  there  can  he  one, 
two,  or  eveyi  three  right  angles;  there  can  he  one,  two,  or  three 
ohtuse  angles. 

EXERCISES 

1.  Show  that  a  triangle  on  the  earth's  surface  whose  sides 
are  the  equator  and  two  meridians,  has  two  of  its  angles  right 
angles,  and  two  of  its  sides  quadrants. 

2.  If,  as  in  Ex.  1,  two  of  the  angles  of  a  spherical  triangle 
are  right  angles,  between  what  limits  must  the  third  angle  lie  ? 


304 


THE  SPHERE 


[Vni,  §  369 


369.  Birectangular    and    Trirectangular    Triangles.      A 

spherical  triangle  having  two  right  angles  is  called  a  birectangu- 


FiG.  250 

lar  spherical  triangle.  A  spherical  triangle  having  all  of  its 
angles  right  angles  is  called  a  trirectangular  spherical  triangle. 
If  Z  P  in  the  figure  is  either  acute  or  obtuse,  while  A  A  and 
B  are  right,  AABP  is  birectangular;  if  ZP  is  also  a  right 
angle,  A  ABP  is  trirectangular,  as  in  Fig.  230,  p.  284, 


EXERCISES 

1.  The  sides  of  a  epherical  triangle  are  80°,  and  126°,  and 
175°.     How  large  are  the  angles  of  its  polar  triangle  ? 

2.  Show  that  in  a  birectangular  triangle  the  sides  opposite 
the  right  angles  are  quadrants. 

3.  Show  that  three  mutually  perpendicular  planes  through 
the  center  of  a  sphere  divide  its  surface  into  eight  congruent 
trirectangular  triangles. 

4.  Show  that  the  area  of  a  trirectangular  triangle  on  a  sphere 
is  one  eighth  of  the  area  of  the  sphere. 

5.  Show  that  each  of  the  sides  of  a  trirectangular  triangle  is 
a  quadrant.  Hence  show  that  the  polar  of  a  trirectangular  tri- 
angle coincides  with  it. 


VIII,  §371]      TRIANGLES  AND  POLYGONS 


305 


370.  Symmetric  Triangles.  Two  spherical  triangles  are 
symmetric  when  their  parts  are  equal  each  to  each,  but  are 
in  opposite  order.     Thus,  in  the  A  ABC  and  A'B'C  (Fig.  251), 


Fig.  251 

if  angles  A  =  A',  B  =  B',  C=  C,  and  sides  AB  =  A'B',  BG  = 
B'C,  CA  =  G'A'j  but  the  order  of  arrangement  is  opposite  in 
the  two  figures,  the  triangles  are  symmetric. 

In  general,  two  symmetric  triangles  cannot  be  superposed 
and  hence  cannot  be  said  to  be  congruent. 

Thus,  if  A  ABC  is  moved  so  that  side  AB  coincides  with  its 
equal,  A'B'j  in  the  symmetric  A  A'B'C,  then  the  vertices  G  and 
(7  lie  on  opposite  sides  of  A'B'.  In  plane  triangles,  A  ABC 
could  be  revolved  about  ^B  till  it  coincided  with  A  A' B'C; 
but  this  is  in  general  impossible  with  spherical  triangles. 

371.  Corollary.  Two  isosceles  syinmetric  spherical  triangles 
are  congrue7it. 

EXERCISES 

1.  Prove  that  the  base  angles  of  an  isosceles  spherical  tri- 
angle are  equal. 

[Hint.  Draw  an  arc  bisecting  the  vertical  angle,  thus  forming  two 
symmetric  triangles.] 

2.  Show  that  if  two  sides  of  a  spherical  triangle  are  quad- 
rants, the  triangle  is  birectangular. 


306 


THE  SPHERE 


[VIII,  §  372 


372.  Theorem  X.  Tivo  triangles  on  the  same  sphere,  or  07i 
equal  spheres,  are  either  congruent  or  symmetric,  if  two  sides 
and  the  included  angle  of  the  one  are  equal,  respectively,  to  two 
sides  and  the  included  angle  of  the  other. 


/ 


Fig.  252  (a) 


Fig.  252  (b) 


Given  the  spherical  A  ABC  and  A'B'C  on  the  same  sphere 
or  equal  spheres,  having  AB  =  A'B',  AC=A'C',  Z  A  =  Z.  A'. 

To  prove  that  ^  ABC  and  A'B'C  are  either  congruent  or 
else  symmetric. 

Proof.  If  the  equal  parts  of  the  two  triangles  are  in  the 
same  order,  A  ABC  can  be  placed  on  A  A'B'C  as  in  the  corre- 
sponding case  of  plane  triangles.     See  Fig.  252  (a). 

If  the  equal  parts  of  the  two  triangles  are  not  in  the  same 
order,construct  A^'^'O"  symmetric  to  A^'^'C".   (Fig.252  (6).) 

In  A  ^5(7  and  A!B'C',  AC  =  A'C",  AB  =  A'B',  Sind  Z  A 
=  Z  B'A'C".  Since  these  parts  are  arranged  in  the  same  order, 
A  ABC  and  A'B'  C"  are  congruent.  Therefore  spherical  A  ABC 
is  symmetric  to  spherical  A  A'B'C.  Why? 

373.  Theorem  XI.  Two  triangles  on  the  same  sphere,  or  on 
equal  spheres,  are  either  congruent  or  symmetric,  if  two  angles  and 
the  included  side  of  the  one  are  equal,  respectively,  to  two  angles 
and  the  included  side  of  the  other.     [Proceed  as  in  §  372.] 


VIII,  §375]       TRIANGLES  AND  POLYGONS 


307 


374.  Theorem  XII.  Two  triangles  on  the  same  sphere,  or  on 
equal  spheres,  are  either  congruent  or  symmetric,  if  the  three  sides 
of  the  one  are  equal,  respectively,  to  the  three  sides  of  the  other. 

[The  proof  is  left  to  the  student.  ] 

375.  Theorem  XIII.     Two  triangles  on  the  same  sphere,  or 
on  equal  spheres,  are  either  congruent  or  symmetric,  if  the  three ' 
angles  of  the  one  are  equal,  respectively,  to  the  three  angles  of 
the  other. 


Fig.  253 

Outline  of  Proof.  If  ABO  and  A'B'C  are  the  two  given 
spherical  triangles  so  that  ZA  =  ZA',ZB  =  ZB',  ZC  =  AC', 
their  polar  triangles  LMN  and  VM^N^  have  the  three  sides 
of   one  equal   to   the   three  sides  of  the  other,  respectively. 

§366 

Then,  by  §  374,  A  LMN  and  L'M'N'  are  either  congruent  or 
symmetric.  In  either  case,  the  three  angles  of  A  LMN  are 
equal  to  the  three  angles  of  A  VM^N,  respectively  ;  and  there- 
fore the  three  sides  of  A  ABC  are  equal  to  the  three  sides  of 
A^'^'C,  respectively.  §  366 

It  follows,  by  §  374,  that  A  ABC  and  AB'G'  are  either  con- 
gruent or  symmetric. 

Note.  Theorems  analogous  to  those  of  §§  41,  43,  44,  etc.,  may  be 
proved  in  a  manner  similar  to  §§  372-375. 


308  THE  SPHERE  [VHI,  §  375 

EXERCISES 

1.  Show  that  two  trihedral  angles  are  congruent  if  they 
intercept  congruent  triangles  on  the  surfaces  of  two  equal 
spheres  whose  centers  are  at  their  vertices,  respectively. 

2.  If  two  trihedral  angles  intercept  symmetric  spherical  tri- 
angles on  the  surface  of  a  sphere  whose  center  is  at  their  ver- 
tices, respectively,  show  that  the  face  angles  and  the  dihedral 
angles  of  one  trihedral  angle  are  equal  to  those  of  the  other, 
but  taken  in  reversed  order. 

[Such  trihedral  angles  are  called  symmetric] 

3.  Prove  the  following  theorem,  which  states  for  trihedral 
angles  (§  361)  a  theorem  analogous  to  that  of  §  372  : 

Two  trihedral  angles  are  either  congruent  or  symmetric  if  two 
face  angles  and  the  included  dihedral  angle  of  the  one  are  respec- 
tively equal  to  two  face  angles  and  the  included  dihedral  angle  of 
the  other. 

[Hint  :  Consider  the  spherical  triangles  cut  out  by  the  two  trihedral 
angles  on  the  surfaces  of  two  equal  spheres  whose  centers  lie  at  the  ver- 
tices of  the  two  trihedral  angles,  and  apply  §  372.] 

4.  Prove  the  following  theorem,  analogous  to  §  373 : 

Two  trihedral  angles  are  either  congruent  or  symmetric  if  two 
dihedral  angles  and  the  included  face  angle  of  the  one  are  respec- 
tively equal  to  two  dihedral  angles  and  the  included  face  angle  of 
the  other. 

5.  State  and  prove  theorems  for  trihedral  angles  similar  to 
Theorems  XII-XIII,  §  374-375. 

6.  Prove  Theorem  XI,  §  373,  by  first  considering,  as  in 
§  375,  the  polars  of  the  given  triangles,  and  applying  §  372. 

7.  Show  that  any  trirectangular  triangle  on  the  earth's  sur- 
face is  congruent  to  the  trirectangular  triangle  formed  by  the 
equator  and  two  meridians  whose  longitude  differs  by  90°. 


VIII,  §  376] 


AREAS   AND  VOLUMES 


309 


PAET   III.     AEEAS   AND   VOLUMES 

376.  Theorem  XIV.  The  area  of  the  surface  generated  by  a 
straight  line  revolving  about  an  axis  in  its  plane  is  equal  to  the 
product  of  the  projection  of  the  line  on  the  axis  and  the  length  of  the 
circle  whose  radius  is  a  perpendicular  erected  at  the  middle  point 
of  the  line  and  terminated  by  the  axis. 


Given  EF,  the  projection  upon  XF  of  AB  revolving  about 
XY,  and  OF  A.  AB  at  its  mid-point,  and  meeting  XY  at  0. 

To  prove  that  the  area  generated  by  AB  =  EF  x  2  ttOP. 

Proof.     Draw  PD  J.  XY,  and  AG  II  XY 

Since  the  surface  generated  by  AB  is  the  lateral  surface 
of  the  frustum  of  a  cone,  the  area  generated  by  AB  is 

f(: 

Now 

Therefore 

Then 

And  AB  x2  ttPD  =  EF  x  2  ttOP. 

That  is,  the  area  generated  by  AB  is  EF  x  2  ttOP. 

If  AB  meets  XY,  the  surface  generated  is  a  conical  surface 
whose  area  again  =z  EF  x  2  ttOP.  §  320 

If  AB  is  parallel  to  XY,  the  surface  generated  is  a  cylin- 
drical surface  whose  area  again  =  EF  x  2  it  OP.  §  313 


2  irAE  +  2  'jtBF^  -  ^B  X  2  tt  .  M^  +  ^^A 

=  ABx2Tr'PD. 

§321 

A  ABC  ^  APOD. 

§  157 

AB:OP=AC:PD. 

Why? 

ABx  PD  =  AC  X  0P=  EFx  OP. 

Why? 

310 


THE  SPHERE 


[VIII,  §  377 


377.  Theorem  XV.  The  area  of  the  surface  of  a  sphere  is 
equal  to  the  product  of  its  diameter  by  the  circumference  of  a 
great  circle.  C 


Given  a  sphere  generated  by  the  revolution  of  the  semicircle 
ABCDE  about  the  diameter  AOE,  S  being  the  area  of  the 
surface,  r  being  the  radius,  and  d  being  the  diameter. 

To  prove  that  S  =  2  irrd. 

Proof.  Inscribe  in  the  semicircle  half  of  a  regular  polygon 
ABCDE,  of  any  number  of  sides,  and  draw  BF,  CO,  DO,  per- 
pendicular to  AE. 

From  0  draw  0P1.AB.     Then  OP  bisects  AB,  Why? 

and  is  equal   to  each  of   the  Js  drawn  from  0  to  the  equal 


Why? 
§  376 


chords  BC,  CD,  DE. 

Now      the  area  generated  hy  AB  =  AF  x  2  tt  •  OP, 
the  area  generated  hj  BC  =  FO  x  2  tt  •  OP, 
the  area  generated  by  CD  =  OG  X  2  tt  -  OP, 
the  area  generated  hy  DE  =  GE  x  2  tt  •  OP. 
Therefore,  if  JS'  denotes  the  surface  generated  by  the  semi- 
polygon, 

S'  =  (AF+FO-^OG+GE)2TrOP=AEx2  7rOP. 
Let  the  number  of  sides  of  the  semipolygon  be  now  indefi- 
nitely increased. 

Then  OP  has  for  its  limit  r,  the  semipolygon  for  its  limit 
the  semicircle,  and  S'  for  its  limit  S.     Hence,  as  in  §  303, 
S  =  AEx2  7rr. 


378.   Corollary  1. 

to  4  TTJ^. 


The  area  of  the  surface  of  a  sphere  is  equal 


VIII,  §  382] 


AREAS  AND  VOLUMES 


311 


379.  Corollary  2.  The  area  of  the  surface  of  a  sphere  is  equal 
to  the  sum  of  the  areas  of  four  great  circles. 

For  S  =  2rx2  7rr  =  4:7rr^  §378 

and  Trr^  is  the  area  of  a  great  circle. 

380.  Corollary  3.  The  areas  of  the  surfaces  of  two  spheres 
are  to  each  other  as  the  squares  of  their  radii;  or,  as  the  squares 
of  their  diameters. 

381.  Zones.  A  zone  is  a  portion  of  the  surface  of  a  sphere 
bounded  by  the  circumferences  of  two  circles  whose  planes  are 
parallel. 


Fig.  256.     Zones  on  the  Earth's  Surface 

The  circumferences  forming  the  boundary  of  a  zone  are  its 
bases. 

If  the  semicircle  NES  is  revolved  about  NS  as  an  axis,  arc 
AB  will  generate  a  zone,  while  points  A  and  B  will  generate 
the  bases  of  the  zone. 

The  altitude  of  a  zone  is  the  perpendicular  distance  between 
the  planes  of  the  bases. 

382.  Corollary  4.  The  area  of  a  zone  of  a  sphere  is  equal  to 
the  product  of  the  altitude  h  of  the  zone  and  the  circumference  of 
a  great  circle;  or  2Trrh,  where  r  is  the  radius  of  the  sphere. 


312 


THE  SPHERE 


[VIII,  §  383 


383.   Lunes.     A  lune   is  a  portion  of   a  spherical   surface 
bounded    by   two    semicircumferences    of    great    circles ;    as 


\  ^' 


i 


Fig.  257 

ABGDA  (Fig.  257).  The  angle  of  a  lune  is  the  angle  formed 
by  its  bounding  arcs.  Thus  BAD  is  the  angle  of  the  lune 
ABCDA. 

384.  Theorem  XVI.  Tlie  area  of  a  lune  is  to  the  area  of  the 
surface  of  the  sphere  as  the  angle  of  the  lune  is  to  four  right 
angles. 


Fig.  258 


Given  the  lune  PAP'B,  let  L  denote  the  area  of  the  lune,  S 
the  area  of  the  surface  of  the  sphere,  and  a  the  angle  of  the 
lune. 


VIII,  §  386]  AREAS  AND  VOLUMES  313 

To  prove  that  L/S  =  Z  a/4  rt.  A. 

Proof.     With  P  as  a  pole  describe  the  great  circle  ABCD. 

Then  the  arc  AB  measures  Z  a  of  the  lune.  Why  ? 

Therefore  arc  ^B/circle  ABCD  =  Z  a/4  rt.  A. 

If  AB  and  ABCD  are  commensurable,  let  their  common 
measure  be  contained  m  times  in  ^S  and  n  times  in  ABCD. 

Then  arc  ^5/circle  ABCD  =  m/n. 

Therefore  a/4  rt.  A  =  m/n.  §  358 

Pass  arcs  of  great  circles  through  each  point  of  division  of 
ABCD  and  the  poles  P  and  P'. 

These  arcs  will  divide  the  entire  surface  into  w  equal  lunes, 
of  which  PAPB  will  contain  m. 

Therefore  L/S  =  m/n, 

or,  L/S  =  a/4  rt.  A. 

If  AB  and  ABCD  are  incommensurable,  the  theorem  can  be 
proved  as  in  §  130.     The  details  are  left  to  the  student. 

385.  Corollary  1.  Hie  area  of  a  lune  whose  angle  is  1°  is 
4  7rrV360  =  7rrV90. 

386.  Corollary  2.  The  area  of  a  lune  whose  angle  is  k°  is 
4  7rr^k/S60  =  7rr^k/90. 

EXERCISES 

1.  If  the  surface  of  a  sphere  is  10  sq.  ft.,  what  is  the  area 
of  a  lune  whose  angle  is  40°?  What  is  the  radius  of  the 
sphere  ?  Ans.    li  sq.  ft. ;  0.89+  ft. 

2.  Show  that  two  lunes  on  the  same  sphere  or  equal  spheres 
have  the  same  ratio  as  their  angles. 

3.  What  is  the  angle  of  a  lune  which  has  the  same  area  as  a 
trirectangular  triangle  ? 

4.  Show  that  the  area  of  a  lune  is  one  ninetieth  of  the  area 
of  a  great  circle  multiplied  by  the  number  of  degrees  in  the 
angle  of  the  lune. 


314 


THE  SPHERE 


[Vni,  §  387 


387.   Theorem  XVII.     Two   symmetric   triangles   are   equal 


m  area. 

Given  the  two  symmetric 
spherical  triangles  ABC  and 
A^B'O. 

To  prove  that 

AABC^AA'B^a. 

Proof.  Let  P  be  the  pole  of  the 
small  circle  passing  through  the 
points  A,  B,  C,  and  draw  the  great 
circle  arcs  PA,  PB,  and  PC. 

Then    PA  =  PB  =  PC. 


Fig.  259 


Why? 

Now  place  the  two  triangles  diametrically  opposite  to  each 
other  and  draw  the  diameter  POP.     Also  draw  the  great  circle 
arcs  PA',  P'B',  and  PC.     Then  the  triangles  PBC  and  PB'C 
are  symmetrical  and  isosceles  and 
therefore  congruent.     §  371. 

Similarly  APCA^APGA', 
and  APAB^APA'B'. 

That  is,  the  three  parts  of  ABC 
are  respectively  congruent  to  the 
three  parts  of  ABC 

Therefore  A  ABC  =  A  A'B'C. 


*i 


Fig.  260 


388.  Corollary  1.  If  tivo  semi- 
circumferences  of  great  circles  BCB' 
and  AC  A  intersect  on  the  surface 
of  a  hemisphere,  the  sum  of  the  areas  of  the  two  opposite  spheri- 
cal triangles  ACB  and  A'CB'  is  equal  to  the  area  of  a  lune 
whose  angle  is  equal  to  ACB. 


[Hint.     Show  tliat  the  triangle  ABC  is  symmetric  to  the  triangle 
A'B'C.     Hence  show  that  A  ACB  +  A  A'CB'  =  lune  A' CB'C] 


VIII,  §  391] 


AREAS  AND  VOLUMES 


315 


389.  Spherical  Degree.  The  area  of  a  lune  whose  angle  is 
1°  is  4  7rr2/360,  or  7rrV90  (§  385).  Half  this  area,  that  is, 
47rrY720  or  Trr^/lSO,  is  often  taken  as  a  unit  of  area  on  the 
sphere,  and  it  is  called  a  spherical  degree. 

390.  Measure  of  Solid  Angles.  A  trihedral  angle  whose 
vertex  is  at  the  center  of  a  sphere  cuts  out  a  spherical  triangle 
on  the  surface  of  the  sphere.  The  area  of  the  spherical  tri- 
angle, in  spherical  degrees,  is  called  the  measure  of  the  trihedral 
angle. 

Likewise  any  polyhedral  angle  is  measured  by  the  area,  in 
spherical  degrees,  that  it  cuts  out  upon  the  surface  of  a  sphere 
whose  center  is  at  its  vertex. 


EXERCISES 

1.  Show  that  the  area  of  a  lune  whose  angle  is  1°  is  2  spher- 
ical degrees. 

2.  Show  that  the  area  of  the  entire 
sphere  is  720  spherical  degrees 

3.  Show  that  the  area  of  a  birectan- 
gular  triangle  whose  third  angle  is  1° 
is  1  spherical  degree. 

4.  Show  that  the  area  of  a  trirectan- 
gular  triangle  is  90  spherical  degrees, 
or  one  eighth  of  the  entire  surface. 

391.  Spherical  Excess.  The  excess  of  the  sum  of  the 
angles  of  a  spherical  triangle  over  180°  is  called  the  spherical 
excess  of  the  triangle. 

If,  for  example,  the  angles  of  a  spherical  triangle  are  80°, 
100°,  and  125°,  the  spherical  excess  of  the  triangle  is  125°. 

Likewise,  the  spherical  excess  of  any  spherical  polygon  is 
the  excess  of  the  sum  of  its  angles  above  the  sum  of  the  angles 
of  a  plane  polygon  of  the  same  number  of  sides. 


316 


THE  SPHERE 


[Vin,  §  392 


392.  Theorem  XVIII.  The  area  of  a  spherical  triangle  is 
equal  to  the  area  of  a  Uine  luhose  angle  is  half  the  spherical  excess 
of  the  triangle. 


Fig. 261 

Given  the  spherical  A  ABC. 

To  prove  that  A  ABC  is  equal  to  a  lune  whose  angle  is 
i(Z^  +  ^-B  +  Z  (7-180°). 

Proof.     Complete  the  great  circles  by  producing  the  sides  of 
the  A  ABC,  as  in  Eig.  261. 

Since  A  AB'C  and  A'BC  are  symmetric,  they  are  equal  in 
area  §  387 

Therefore       lune  ABA'C=  A  ABC  +  A  AB'C.  §  388 

But,  denoting  the  area  of  the  whole  sphere  by  S, 

A  CB'A  -j-  A  ACB  +  A  ABC-^A  AB'C  =  iJS.     Why  ? 
Therefore 

(lune  BCB'A  -  A  ABC)  +  (lune  CAC'B  -  A  ABC) 

+  lune  ABA'C=  ^  S.  Why  ? 

Therefore,  transposing,  we  obtain 
2  A  ABC  =  lune  .^1B^'C+  lune  BCB'A  +  lune  CAC'B  -  I S. 
But  -1-  S  is  the  area  of  a  lune  whose  angle  is  180°. 
Therefore  A  ABC  is  equal  to  a  lune  whose  angle  is 

i(AA+ZB-{-ZC-  180°).  §  384 


VIII,  §  396]  AREAS  AND  VOLUMES  317 

393.  Corollary  1.  The  area  of  a  spherical  triangle,  measured 
in  spherical  degrees,  is  numericallij  equal  to  its  spherical  excess. 

Note.  This  result  enables  us  to  compute  the  area  of  any  spherical  tri- 
angle in  ordinary  units  of  area,  when  we  know  its  angles  and  the  radius 
of  the  sphere.  Thus,  if  r  denotes  the  radius  of  the  sphere,  E  the  spheri- 
cal excess,  and  A  the  required  area,  we  have,  by  §  385 

^-^""l^Q-  180* 

394.  Corollary  2.  The  area  of  a  trirectangular  triangle  is 
90  spherical  degrees. 

395.  Corollary  3.  Tlie  area  of  any  spherical  triaiigle  is  to  the 
area  of  the  entire  sphere  as  its  spherical  excess  is  to  720°. 

396.  Solid  Angles.  In  general,  if  any  closed  polygon  or  curve 
is  drawn  on  the  surface  of  a  sphere,  the  figure  formed  by  all 
radii  of  the  sphere  that  join  the  center  to  the  points  of  this 
figure  on  the  spherical  surface  is  called  a  solid  angle.  The  area 
on  the  surface  of  the  sphere  cut  out  by  such  a  solid  angle,  in 
spherical  degrees,  is  the  measure  of  the  solid  angle. 

EXERCISES 

1.  What  is  the  measure  of  a  hemisphere  in  spherical  degrees  ? 

2.  The  radius  of  a  sphere  is  2  ft.  Find  the  area  of  a  triangle 
on  its  surface  whose  angles  are  75°,  35°,  105°,  respectively. 
Solve  first  by  §  392  ;  then  by  §  394.  Ans.    7  7r/9  sq.  ft. 

3.  The  radius  of  the  earth  is  approximately  4000  miles. 
Find  the  entire  area.  Show  that  the  area  in  square  miles  of 
one  spherical  degree  is  approximately  278,000  square  miles. 

4.  Find  how  large  a  triangle  on  the  earth's  surface  would 
have  the  total  sum  of  its  three  angles  equal  to  181°. 

5.  Show  that  a  region  containing  about  270,000  sq.  mi.  on 
the  earth  contains  no  triangle  whose  spherical  excess  is  1°. 

6.  What  is  the  area  of  the  state  in  which  you  live  ?  What 
is  its  measure  in  spherical  degrees  ? 


318 


THE  SPHERE 


[VIII,  §  397 


397.   Theorem  XIX.     The  volume   V  of  a  sphere  is  equal 
to  the  product  of  its  surface  by  one  third  of  its  radius ;  or, 


•  .'i? 

A            1 

A. 

B 

Fig.  2G2 

Given  a  sphere  whose  center  is  O;  let  S  denote  its  surface, 
r  its  radius,  and  V  its  volume. 

To  prove  that  V=  S  x  r/3  =  4  ttt^/S. 

Proof.  Circumscribe  about  the  sphere  any  polyhedron  as 
D-ABC,  and  denote  its  surface  by  S'  and  its  volume  by  V\ 

Form  pyramids,  as  0-ABC,  etc.,  having  the  faces  of  the 
polyhedron  as  bases  and  the  center  of  the  sphere  as  a  common 
vertex. 

These  pyramids  will  have  a  common  altitude  equal  to  r, 
and  the  volume  of  each  pyramid  is  equal  to  its  base  X  r/3. 

Why? 

Therefore  V  ==S'x  r/3.  Why  ? 

If  the  number  of  pyramids  is  indefinitely  increased  by  pass- 
ing planes  tangent  to  the  sphere  at  points  where  the  edges  of 
the  pyramids  cut  the  surface  of  the  sphere,  as  in  Fig.  262,  the 
difference  between  S  and  S'  becomes  as  small  as  we  please ;  the 
difference  between  V  and  V  becomes  as  small  as  we  please. 


VIII,  §  400]  AREAS  AND  VOLUMES  319 

But  however  great  the  number  of  pyramids, 

r=/S'Xr/3. 
Therefore,  as  in  §  303,  V=Sx  r/3. 

Since  /S  =  4  Tr?'^, 

it  follows  that     F  =  4  Trr^  x  r/3  =  4  7r?V3. 

398.  Corollary  1.  The  volumes  of  two  spheres  are  to  each 
other  as  the  cubes  of  their  radii,  or  as  the  cubes  of  their  diameters. 

399.  Corollary  2.  The  volmne  of  the  pyramidal  piece  cut 
out  of  a  sphere  by  any  polyhedral  angle  whose  vertex  is  at  the 
center  is  equal  to  one  third  the  area  of  the  spherical  polygon  cut 
out  of  the  surface  times  the  radius. 

400.  Corollary  3.  The  prismoid  formula  (§  336)  holds  for 
a  sphere. 

[Hint.  Two  parallel  planes  that  include  the  entire  sphere  are  tan- 
gent planes  at  the  ends  of  a  diameter ;  these  cut  the  sphere  in  only  one 
point  each.  A  plane  parallel  to  these  two  and  halfway  between  thera 
cuts  the  sphere  in  a  great  circle.  Hence,  in  the  notation  of  §  336,  B  —  0^ 
T  =  0,  ilf  =  4  Trr^,  h  =  2r  ;  hence  the  prismoid  formula  would  give 
V=h  r^+  r  +  4Jf-1  ^  2  r  rO  +  0  +  4  Trr^-i  ^Airr^^ 

which,  by  §  397,  is  correct.] 

[Note,  As  a  matter  of  fact,  the  prismoid  formula  holds  for  the  por- 
tion of  a  sphere  intercepted  between  any  two  parallel  planes.] 

EXERCISES 

1.  Assuming  that  the  earth  is  a  sphere  whose  radius  is 
4000  mi.,  find  its  volume. 

2.  Show  that  a  cube  circumscribed  about  a  sphere  has  a 
volume  8  r^.  Hence  show  that  the  sphere  occupies  a  little 
more  than  half  this  volume 


320  THE  SPHERE  [VIH,  §  400 

3.  Find  the  volume  of  the  material  in  a  hollow  sphere,  if 
the  radius  of  the  outer  surface  is  6  in.  and  that  of  the  inner 
surface  is  5  in. 

4.  Show  that  the  volume  of  a  hollow  sphere  whose  outer 
and  inner  radii  are  B  and  r,  respectively,  is  4  tt  {R^  —  r^)/3. 

5.  Find  the  volume  of  the  material  in  a  hollow  sphere 
whose  outer  radius  is  10  in.,  if  the  material  is  ^  in.  thick. 

6.  Show  that  the  volume  of  a  sphere  in  terms  of  its  diame- 
ter, d,  is  7rdy6. 

7.  If  the  radius  of  one  sphere  is  twice  that  of  another,  how 
do  their  volumes  compare  ? 

8.  If  the  volume  of  one  sphere  is  twice  that  of  another,  how 
do  their  radii  compare  ? 

9.  Find  approximately  the  radius  of  a  sphere  whose  volume 
is  100  cu.  in. 

10.  How  many  shot  -^^  in.  in  diameter  can  be  made  from 
10  cu.  in.  of  lead  ? 

11.  If  oranges  3  in.  in  diameter  sell  for  30  cents  per  dozen, 
and  those  4  in.  in  diameter  sell  for  50  cents  per  dozen,  which 
are  the  cheaper  by  volume  ? 

12.  If  the  skins  are  of  equal  thickness,  which  of  the  oranges 
of  Ex.  11  has  the  greater  percentage  of  skin  to  the  cubic  inch 
of  volume  ? 

13.  Assuming  that  raindrops  are  practically  spherical,  if 
the  diameter  of  one  drop  is  half  that  of  another,  how  do  their 
volumes  compare  ?   their  areas  ? 

14.  Which  of  the  two  drops  of  Ex.  13  has  the  greater  ratio 
of  area  to  volume  ?  How  much  greater  ?  Which  will  fall  the 
more  rapidly  through  the  air  ? 

[Hint.  The  greater  the  ratio  of  area  to  volume,  for  the  same  material, 
the  slower  the  body  will  fall  through  the  air.  ] 


VIII,  §400]  AREAS  AND  VOLUMES  321 

15.  Explain,  by  the  principle  of  Ex.  14,  why  very  small  dust 
particles  remain  floating  in  the  air  for  a  long  time. 

16.  The  same  amount  of  material,  in  the  form  of  a  cube,  is 
melted  and  cast  into  a  sphere.  Is  the  surface  area  less  in  the 
form  of  the  cube  or  in  that  of  the  sphere  ? 

[Hint.  Assume  the  cube  to  be  1  unit  on  each  edge ;  find  the  radius 
of  the  resulting  sphere,] 

17.  If  a  surveyor  wishes  to  be  certain  that  the  sum  of  the 
angles  in  any  triangle  in  a  region  on  the  earth's  surface  shall 
be  equal  to  180°  to  within  one  minute,  how  large  may  the  re- 
gion be  ?  Ans.  About  4600  sq.  mi. 

18.  Demonstrate  the  existence  of  spherical  triangles  with 
three  obtuse  angles  from  the  existence  of  triangles  whose  sides 
are  very  short. 


IT 


TABLES 


TABLE   I 

Ratios  of  the  Sides  of  Right  Triangles 

and 

Chords  and  Arcs  of  a  Unit  Circle 

TABLE   II 

Squares  and  Square  Roots  of  Numbers 
Cubes  and  Cube  Roots  of  Numbers 

TABLE   III 

Values  of  Important  Numbers 

including 

Units  of  Measurement 


TABLE   I 

RATIOS  OF   THE   SIDES  OF  RIGHT   TRIANGLES 

AND 

LENGTHS  OF  CHORDS  AND  ARCS  OF  A  UNIT  CIRCLE 

EXPLANATION   OF   TABLE   I 

1.  Ratios  of  the  Sides  of  Right  Triangles.  If  an  angle 
given  in  the  Angle  Column  is  one  acute  angle  of  a  right  triangle: 

The  Sine  Column  gives  the  ratio  of  the  side  opposite  the  angle 
to  the  hypotenuse ; 

The  Tangent  Column  gives  the  ratio  of  the  side  opposite  the 
angle  to  the  side  adjacent  to  the  angle. 

To  find  the  Cosine  of  any  angle,  take  the  sine  of  the  comple- 
ment of  that  angle. 

2.  Chords  and  Arcs  of  a  Unit  Circle.  If  an  angle  given  in 
the  Angle  Column  is  an  angle  at  the  center  of  a  circle  of  unit 
radius : 

The  Chord  Column  gives  the  length  of  the  chord  that  subtends 
that  angle ; 

The  Arc  Column  gives  the  length  of  the  arc  that  subtends  that 
angle. 

To  find  the  lengths  of  chords  or  arcs  of  any  circle  of  radius  r, 
multiply  the  values  given  in  the  table  by  that  radius. 

The  table  is  limited  to  angles  less  than  90° ;  but  to  find  the 
chord  that  subtends  an  obtuse  angle,  first  take  half  the  angle,  find 
the  sine  of  this  half  angle,  and  multiply  by  2.  This  follows 
from  the  fact  that  the  chord  of  any  angle  is  twice  the  sine  of 
half  that  angle. 


I] 

Quantities  Determined  by  a  Given  Angle 

iii 

Angle 

Sine 

Tan- 
gent 

Chord 

Arc 

Angle 

Sine 

Tan- 
gent 

Chord 

Arc 

0°00' 

.0000 

.0000 

.0000 

.0000 

9°  00' 

.1564 

.1584 

.1569 

.1571 

10 

.0029 

.0029 

.0029 

.0029 

10 

.1593 

.1614 

.1598 

.1600 

20 

.0058 

.0058 

.0058 

.0058 

20 

.1622 

.1644 

.1627 

.1629 

30 

.0087 

.0087 

.0087 

.0087 

30 

.1650 

.1673 

.1656 

.1658 

40 

.0116 

.0116 

.0116 

.0116 

40 

.1679 

.1703 

.1685 

.1687 

50 

.0145 

.0145 

.0145 

.0145 

50 

.1708 

.1733 

.1714 

.1716 

1°00' 

.0175 

.0175 

.0175 

.0175 

10°  00' 

.1736 

.1763 

.1743 

.1745 

10 

.0204 

.0204 

.0204 

.0204 

10 

.1765 

.1793 

.1772 

.1774 

20 

.0233 

.0233 

.0233 

.0233 

20 

.1794 

.1823 

.1801 

.1804 

30 

.0262 

.0262 

.0262 

.0262 

30 

.1822 

.1853 

.1830 

.1833 

40 

.0291 

.0291 

.0291 

.0291 

40 

.1851 

.1883 

.1859 

.1862 

50 

.0320 

.0320 

.0320 

.0320 

50 

.1880 

.1914 

.1888 

.1891 

2°  00' 

.0349 

.0349 

.0349 

.0349 

11°00' 

.1908 

.1944 

.1917 

.1920 

10 

.0378 

.0378 

.0378 

.0378 

10 

.1937 

.1974 

.1946 

.1949 

20 

.0407 

.0407 

.0407 

.0407 

20 

.1965 

.2004 

.1975 

.1978 

30 

.0436 

.0437 

.0436 

.0436 

30 

.19^)4 

.2035 

.2004 

.2007 

40 

.0465 

.0466 

.0465 

.0465 

40 

.2022 

.2065 

.2033 

.2036 

50 

.0494 

.0495 

.0494 

.0495 

50 

.2051 

.2095 

.2062 

.2065 

3°  00' 

.0523 

.0524 

.0524 

.0524 

12°  00' 

.2079 

.2126 

.2091 

.2094 

10 

.0552 

.0553 

.0553 

.0553 

10 

.2108 

.2156 

.2119 

.2123 

20 

.0581 

.0582 

.0582 

.0582 

20 

.2136 

.2186 

.2148 

.2153 

30 

.0610 

.0612 

.0611 

.0611 

30 

.2164 

.2217 

.2177 

.2182 

40 

.0640 

.0641 

.0640 

.0640 

40 

.2193 

.2247 

.2206 

.2211 

50 

.0669 

.0670 

.0669 

.0669 

50 

.2221 

.2278 

.2235 

.2240 

4°  00' 

.0698 

.0699 

.0698 

.0698 

13°  00' 

.2250 

.2309 

.2264 

.2269 

10 

.0727 

.0729 

.0727 

.0727 

10 

.2278 

.2339 

.2293 

.2298 

20 

.0756 

.0758 

.0756 

.0756 

20 

.2306 

.2370 

.2322 

.2327 

30 

.0785 

.0787 

.0785 

.0785 

30 

.2334 

.2401 

.2351 

.2356 

40 

.0814 

.0816 

.0814 

.0814 

40 

.2363 

.2432 

.2380 

.2385 

50 

.0843 

.0846 

.0843 

.0844 

50 

.2391 

.2462 

.2409 

.2414 

6°  00' 

.0872 

.0875 

.0872 

.0873 

14°  00' 

.2419 

.2493 

.2437 

.2443 

10 

.0901 

.0904 

.0901 

.0902 

10 

.2447 

.2524 

.2466 

.2473 

20 

.0929 

.0934 

.0931 

.0931 

20 

.2476 

.2555 

.2495 

.2502 

30 

.0958 

.0963 

.0960 

.0960 

30 

.2504 

.2586 

.2524 

.2531 

40 

.0987 

.0992 

.0989 

.0989 

40 

.2532 

.2617 

.2553 

.2560 

50 

.1016 

.1022 

.1018 

.1018 

50 

.2560 

.2648 

.2582 

.2589 

6°  00' 

.1045 

.1051 

.1047 

.1047 

15°00' 

.2588 

.2679 

.2611 

.2618 

10 

.1074 

.1080 

.1076 

.1076 

10 

.2616 

.2711 

.2639 

.2647 

20 

.1103 

.1110 

.1105 

.1105 

20 

.2644 

.2742 

.2668 

.2676 

30 

.1132 

.1139 

.1134 

.1134 

30 

.2672 

.2773 

.2697 

.2705 

40 

.1161 

.1169 

.1163 

.1164 

40 

.2700 

.2805 

.2726 

.2734 

50 

.1190 

.1198 

.1192 

.1193 

50 

.2728 

.2836 

.2755 

.2763 

7°  00' 

.1219 

.1228 

.1221 

.1222 

16°  00' 

.2756 

.2867 

.2783 

.2793 

10 

.1248 

.1257 

.1250 

.1251 

10 

.2784 

.2899 

.2812 

.2822 

20 

.1276 

.1287 

.1279 

.1280 

20 

.2812 

.2931 

.2841 

.2851 

30 

.1305 

.1317 

.1308 

.1309 

30 

.2840 

.2962 

.2870 

.2880 

40 

.1334 

.1346 

.1337 

.1338 

40 

.2868 

.2994 

.2899 

.2909 

50 

.1363 

.1376 

.1366 

.1367 

50 

.2896 

.3026 

.2927 

.2938 

8°  00' 

.1392 

.1405 

.1395 

.1396 

17°  00' 

.2924 

.3057 

.2956 

.2967 

10 

.1421 

.1435 

.1424 

.1425 

10 

.2952 

.3089 

.2985 

.2996 

20 

.1449 

.1465 

.1453 

.1454 

20 

.2979 

.3121 

.3014 

.3025 

30 

.1478 

.1495 

.1482 

.1484 

30 

.3007 

.3153 

.3042 

.3054 

40 

.1507 

.1524 

.1511 

.1513 

40 

.3035 

.3185 

.3071 

.3083 

50 

.1536 

.1554 

.1540 

.1542 

50 

.3062 

.3217 

.3100 

.3113 

9°  00' 

.1564 

.1584 

.1569 

.1571 

18°  00' 

.3090 

.3249 

.3129 

.3142 

IV 

Quantities 

Determined  by  a  Given  Angle 

[I 

Angle 

Sine 

Tan- 
gent 

Chord 

Arc 

Angle 

Sine 

Tan- 
gent 

Chord 

Arc 

18°00' 

.3090 

.3249 

.3129 

.3142 

27°  00' 

.4540 

.5095 

.4669 

.4712 

10 

.3118 

.3281 

.3157 

.3171 

10 

.4566 

.5132 

.4697 

.4741 

20 

.3145 

.3314 

.3186 

.3200 

20 

.4592 

.5169 

.4725 

.4771 

30 

.3173 

.3346 

.3215 

.3229 

30 

.4617 

.5206 

.4754 

.4800 

40 

.3201 

.3378 

.3244 

.3258 

40 

.4643 

.5243 

.4782 

.4829 

50 

.3228 

.3411 

.3272 

.3287 

50 

.4669 

.5280 

.4810 

.4858 

19°  00' 

.3256 

.3443 

.3301 

.3316 

28°  00' 

.4695 

.5317 

.4838 

.4887 

10 

.3283 

.3476 

.3330 

.3345 

10 

.4720 

.5354 

.4867 

.4916 

20 

.3311 

.3508 

.3358 

.3374 

20 

.4746 

.5392 

.4895 

.4945 

30 

.3338 

.3541 

.3387 

.3403 

30 

.4772 

.5430 

.4923 

.4974 

40 

.3365 

.3574 

.3416 

.3432 

40 

.4797 

.5467 

.4951 

.5003 

50 

.3393 

.3607 

•3444 

.3462 

50 

.4823 

.5505 

.4979 

.5032 

20°  00' 

.3420 

.3640 

.3473 

.3491 

29°  00' 

.4848 

.5543 

.5008 

.5061 

10 

.3448 

.3673 

.3502 

.3520 

10 

.4874 

.5581 

.5036 

.5091 

20 

.3475 

.3706 

.3530 

.3549 

20 

.4899 

.5619 

.5064 

.5120 

30 

.3502 

.3739 

.3559 

.3578 

30 

.4924 

.5658 

.5092 

.5149 

40 

.3529 

.3772 

.3587 

.3607 

40 

.4950 

.5696 

.5120 

.5178 

50 

.3557 

.3805 

.3616 

.3636 

50 

.4975 

.5735 

.5148 

.5207 

21°  00' 

.3584 

.3839 

.3645 

.3665 

30°  00' 

.5000 

.5774 

.5176 

.5236 

10 

.3611 

.3872 

.3673 

.3694 

10 

.5025 

.5812 

.5204 

.5265 

20 

.3638 

.3906 

.3702 

.3723 

20 

.5050 

.5851 

.5233 

.5294 

30 

.3665 

.3939 

.3730 

.3752 

30 

.5075 

.5890 

.5261 

.5323 

40 

.3692 

.3973 

.3759 

.3782 

40 

.5100 

.5930 

.5289 

.5352 

50 

.3719 

.4006 

.3788 

.3811 

50 

.5125 

.5969 

.5317 

.5381 

22°  00' 

.3746 

.4040 

.3816 

.3840 

31°  00' 

.5150 

.6009 

.5345 

.5411 

10 

.3773 

.4074 

.3845 

.3869 

10 

.5175 

.6048 

.5373 

.5440 

20 

.3800 

.4108 

.3873 

.3898 

20 

.5200 

.6088 

.5401 

.5469 

30 

.3827 

.4142 

.3902 

.3927 

30 

.5225 

.6128 

.5429 

.5498 

40 

.3854 

.4176 

.3930 

.3956 

40 

.5250 

.6168 

.5457 

.5527  . 

50 

.3881 

.4210 

.3959 

.3985 

50 

.5275 

.6208 

.5485 

.5556 

23° 00' 

.3907 

.4245 

.3987 

.4014 

32°  00' 

.5299 

.6249 

.5513 

.5585 

10 

.3934 

.4279 

.4016 

.4043 

10 

.5324 

.6289 

.5M1 

.5614 

20 

.3961 

.4314 

.4044 

.4072 

20 

.5348 

.6330 

.5569 

.5643 

30 

.3987 

.4348 

.4073 

.4102 

30 

.5373 

.6371 

.5597 

.5672 

40 

.4014 

.4383 

.4x01 

.4131 

40 

.5398 

.6412 

.5625 

.5701 

50 

.4041 

.4417 

.4130 

.4160 

50 

.5422 

.6453 

.5652 

.5730 

24° 00' 

.4067 

.4452 

.4158 

.4189 

33° 00' 

.5446 

.6494 

.5680 

.5760 

10 

.4094 

.4487 

.4187 

.4218 

10 

.5471 

.6536 

.5708 

.5789 

20 

.4120 

.4522 

.4215 

.4247 

20 

.5495 

.6577 

.5736 

.5818 

30 

.4147 

.4557 

.4244 

.4276 

30 

.5519 

.6619 

.5764 

.5847 

40 

.4173 

.4592 

.4272 

.4:^5 

40 

.5544 

.6661 

.5792 

.6876 

50 

.4200 

.4628 

.4300 

.4334 

50 

.5568 

.6703 

.5820 

.5905 

25°  00' 

.4226 

.4663 

.4329 

.4363 

34°  00' 

.5592 

.6745 

.5847 

.5934 

10 

.4253 

.4699 

.4357 

.4392 

10 

.5616 

.6787 

.5875 

.5963 

20 

.4279 

.4734 

.4386 

.4422 

20 

.5640 

.6830 

.5903 

.5992 

30 

.4305 

.4770 

.4414 

.4451 

30 

.5664 

.6873 

.5931 

.6021 

40 

.4331 

.4806 

.4442 

.4480 

40 

.5688 

.6916 

.5959 

.6050 

50 

.4358 

.4841 

.4471 

.4509 

50 

.5712 

.6959 

.5986 

.6080 

26°  00' 

.4384 

.4877 

.44^)9 

.4538 

35° 00' 

.5736 

.7002 

.6014 

.6109 

10 

.4410 

.4913 

.4527 

.4567 

10 

.57(50 

.7046 

.6042 

.61.38 

20 

.4436 

.4950 

.4556 

.4596 

20 

.5783 

.7089 

.6070 

.6167 

30 

.4462 

.498() 

.4584 

.4625 

30 

.5807 

.7133 

.6097 

.6196 

40 

.4488 

.5022 

.4612 

.4654 

40 

.5831 

.7177 

.6125 

.6225 

50 

.4514 

.5059 

.4641 

.4683 

50 

.5854 

.7221 

.6153 

.6254 

27°  00' 

.4540 

.5095 

.4669 

.4712 

36°  00' 

.5878 

.7265 

.6180 

.6283 

1] 

Quantities 

Determined  by  a  Giyen  Angle 

V 

Angle 

Sine 

Tan- 
gent 

Chord 

Arc 

Angle 

Sine 

Tan- 
gent 

Chord 

1 

Arc 

36°  00' 

.5878 

.7265 

.6180 

.6283 

45°  00' 

.7071 

1.0000 

.7654 

.7854 

10 

.5901 

.7310 

.6208 

.6312 

10 

.7092 

1.0058 

.7681 

.7883 

20 

.5925 

.7355 

.6236 

.6341 

*  20 

.7112 

1.0117 

.7707 

.7912 

30 

.5948 

.7400 

.6263 

.6370 

30 

.7133 

1.0176 

.7734 

.7941 

40 

.5972 

.7445 

.6291 

.6400 

40 

.7153 

1.0235 

.7761 

.7970 

50 

.5995 

.7490 

.6318 

.6429 

50 

.7173 

1.0295 

.7788 

.7999 

37°  00' 

.6018 

.7536 

.6346 

.6458 

46°  00' 

.7193 

1.0355 

.7815 

.8029 

10 

.6041 

.7581 

.6374 

.6487 

10 

.7214 

1.0416 

.7841 

.8058 

20 

.6065 

.7627 

.6401 

.6516 

20 

.7234 

1.0477 

.7868 

.8087 

30 

.6088 

.7673 

.6429 

.6545 

30 

.7254 

1.0538 

.7895 

.8116 

40 

.6111 

.7720 

.6456 

.6574 

40 

.7274 

1.0599 

.7922 

.8145 

50 

.6134 

.7766 

.6484 

.6603 

50 

.7294 

1.0661 

.7948 

.8174 

38°  00' 

.6157 

.7813 

.6511 

.6632 

47°  00' 

.7314 

1.0724 

.7975 

.8203 

10 

.6180 

.7860 

.6539 

.6661 

10 

.7333 

1.0786 

.8002 

.8232 

20 

.6202 

.7907 

.6566 

.6690 

20 

.7353 

1.0850 

.8028 

.8261 

30 

.6225 

.7954 

.6594 

.6720 

30 

.7373 

1.0913 

.8055 

.8290 

40 

.6248 

.8002 

.6621 

.6749 

40 

.7392 

1.0977 

.8082 

.8319 

50 

.6271 

.8050 

.6649 

.6778 

50 

.7412 

1.1041 

.8108 

.8348 

39°  00' 

.6293 

.8098 

.6676 

.6807 

48°  00' 

.7431 

1.1106 

.8135 

.8378 

10 

.6316 

.8146 

.6704 

.6836 

10 

.7451 

1.1171 

.8161 

.8407 

20 

.6338 

.8195 

.6731 

.6865 

20 

.7470 

1.1237 

.8188 

.8436 

30 

.6361 

.8243 

.6758 

.6894 

30 

.7490 

1.1303 

.8214 

.8465 

40 

.6383 

.8292 

.6786 

.6923 

40 

.7509 

1.1369 

.8241 

.8494 

50 

.6406 

.8342 

.6813 

.6952 

50 

.7528 

1.1436 

.8267 

.8523 

40°  00' 

.6428 

.8391 

.6840 

.6981 

49°  00' 

.7547 

1.1504 

.8294 

.8552 

10 

.6450 

.8441 

.6868 

.7010 

10 

.7566 

1.1571 

.8320 

.8581 

20 

.6472 

.8491 

.6895 

.7039 

20 

.7585 

1.1640 

.8347 

.8610 

30 

.6494 

.8541 

.6922 

.7069 

30 

.7604 

1.1708 

.8373 

.8639 

40 

.6517 

.8591 

.69.50 

.7098 

40 

.7623 

1.1778 

.8400 

.8668 

50 

.6539 

.8642 

.6977 

.7127 

50 

.7642 

1.1847 

.8426 

.8698 

41°  00' 

.6561 

.8693 

.7004 

.7156 

50°  00' 

.7660 

1.1918 

.8452 

.8727 

10 

.6583 

.8744 

.7031 

.7185 

10 

.7679 

1.1988 

.8479 

.8756 

20 

.6604 

.87% 

.7059 

.7214 

20 

.7698 

1.20.59 

.8505 

.8785 

30 

.6626 

.8847 

.7086 

.7243 

30 

.7716 

1.2131 

.8531 

.8814 

40 

.6648 

.8899 

.7113 

.7272 

40 

.7735 

1.2203 

.8558 

.8843 

50 

.6670 

.8952 

.7140 

.7301 

50 

.7753 

1.2276 

.8584 

.8872 

42°  00' 

.6691 

.9004 

.7167 

.7330 

51°  00' 

.7771 

1.2349 

.8610 

.8901 

10 

.6713 

.9057 

.7195 

.7359 

10 

.7790 

1.2423 

.8636 

.8930 

20 

.6734 

.9110 

.7222 

.7389 

20 

.7808 

1.2497 

.8663 

.8959 

30 

.6756 

.9163 

.7249 

.7418 

30 

.7826 

1.2572 

.8689 

.8988 

40 

.6777 

.9217 

.7276 

.7447 

40 

.7844 

1.2647 

.8715 

.9018 

50 

.6799 

.9271 

.7303 

.7476 

50 

.7862 

1.2723 

.8741 

.9047 

43°  00' 

.6820 

.9325 

.7330 

.7505 

52°  00' 

.7880 

1.2799 

.8767 

.9076 

10 

.6841 

.9380 

.7357 

.7534 

10 

.7898 

1.2876 

.8794 

.9105 

20 

.6862 

.9435 

.7384 

.7563 

20 

.7916 

1.2954 

.8820 

.9134 

30 

.6884 

.9490 

.7411 

.7592 

30 

.7934 

1.3032 

.8846 

.9163 

40 

.6905 

.9545 

.7438 

.7621 

40 

.7951 

1.3111 

.8872 

.9192 

50 

.6926 

.9(301 

.7465 

.7650 

'  50 

.7969 

1.3190 

.8898 

.9221 

44°  00' 

.6947 

.9657 

.7492 

.7679 

53°  00' 

.7986 

1.3270 

.8924 

.9250 

10 

.6967 

.9713 

.7519 

.7709 

10 

.8004 

1.3351 

.8950 

.9279 

20 

.6988 

.9770 

.7546 

.7738 

20 

.8021 

1.3432 

.8976 

.9308 

30 

.7009 

.9827 

.7573 

.7767 

30 

.8039 

1.3514 

.9002 

.9338 

40 

.7030 

.9884 

.7600 

.7796 

40 

.8056 

1.3597 

.9028 

.9367 

50 

.7050 

.9942 

.7627 

.7825 

50 

.8073 

1.3680 

.9054 

.93^)6 

45°  00' 

.7071 

1.0000 

.7654 

.7854 

54°  00' 

.8090 

1.3764 

.9080 

.9425 

vi 

Quantities  Determined  by  i 

a  Given  Angle 

[1 

Angle 

Sine 

Tan- 
gent 

Chord 

- 

Angle 

Sine 

Tan- 
gent 

Chord 

Arc 

64°  00' 

.801X) 

1.3764 

.9080 

.9425 

63°  00' 

.8910 

1.9626 

1.0450 

1.0996 

10 

.8107 

1.3848 

.9106 

M64: 

10 

.8923 

1.9768 

1.0475 

1.1025 

20 

.8124 

1.3934 

.9132 

.9483 

20 

.8936 

1.9912 

1.0500 

1.1054 

30 

.8141 

1.4019 

.9157 

.9512 

30 

.8949 

2.0057 

1.0524 

1.1083 

40 

.8158 

1.4106 

.9183 

.9541 

40 

.8962 

2.0204 

1.0549 

1.1112 

50 

.8175 

1.4193 

.9209 

.9570 

50 

.8975 

2.0353 

1.0574 

1.1141 

65^00' 

.8192 

1.4281 

.9235 

.9599 

64°  00' 

.8988 

2.0503 

1.0598 

1.1170 

10 

.8208 

1.4370 

.9261 

.9628 

10 

.9001 

2.0(>55 

1.0623 

1.1199 

20 

.8225 

1.4460 

.9287 

.9657 

20 

.9013 

2.0809 

1.0648 

1.1228 

30 

.8241 

1.4550 

.9312 

.9687 

30 

.9026 

2.0965 

1.0672 

1.1257 

40 

.8258 

1.4641 

.9338 

.9716 

40 

.9038 

2.1123 

1.0697 

1.1286 

50 

.8274 

1.4733 

.9364 

.9745 

50 

.9051 

2.1283 

1.0721 

1.1316 

56°  00' 

.8290 

1.4826 

.9389 

.9774 

65°  00' 

.9063 

2.1445 

1.0746 

1.1345 

10 

.8307 

1.4919 

.9415 

.9803 

10 

.9075 

2.1609 

1.0771 

1.1374 

20 

.8323 

1.5013 

.9441 

.9832 

20 

.9088 

2.1775 

1.0795 

1.1403 

30 

.8339 

1.5108 

.9466 

.9861 

30 

.9100 

2.1t)43 

1.0820 

1.1432 

40 

.8355 

1.5204 

.9492 

.9890 

40 

.9112 

2.2113 

1.0844 

1.1461 

50 

.8371 

1.5301 

.9518 

.9919 

50 

.9124 

2.2286 

1.0868 

1.1490 

67° 00' 

.8387 

1.5399 

.9543 

.9948 

66°  00' 

.9135 

2.2460 

1.C893 

1.1519 

10 

.8403 

1.5497 

.9569 

.9977 

10 

.9147 

2.2637 

1.0917 

1.1548 

20 

.8418 

1.5597 

.9594 

1.0007 

20 

.9159 

2.2817 

1.0<)42 

1.1577 

30 

.8434 

1.5697 

.9620 

1.0036 

30 

.9171 

2.2998 

1.09()6 

1.1606 

40 

.8450 

1.5798 

.9645 

1.0065 

40 

.9182 

2.3183 

1.0990 

1.1636 

50 

.8465 

1.5900 

.9671 

1.0094 

50 

.9194 

2.3369 

1.1014 

1.1665 

68°  00' 

.8480 

1.6003 

.9696 

1.0123 

67°  00' 

.9205 

2.3559 

1.1039 

1.1694 

10 

.8496 

1.6107 

.9722 

1.0152 

10 

.9216 

2.3750 

1.10(33 

1.1723 

20 

.8511 

1.6212 

.9747 

1.0181 

20 

.9228 

2.3945 

1.1087 

1.1752 

30 

.8526 

1.6319 

.9772 

1.0210 

30 

.9239 

2.4142 

1.1111 

1.1781 

40 

.8542 

1.6426 

.9798 

1.0239 

40 

.9250 

2.4342 

1.1136 

1.1810 

50 

.8557 

1.6534 

.9823 

1.0268 

50 

.9261 

2.4545 

1.1160 

1.1839 

59°00' 

.8572 

1.6643 

.9848 

1.0297 

68°  00' 

.9272 

2.4751 

1.1184 

1.1868 

10 

.8587 

1.6753 

.9874 

1.0327 

10 

.9283 

2.4960 

1.1208 

1.1897 

20 

.8601 

1.6864 

.9899 

1.0356 

20 

.9293 

2.5172 

1.1232 

1.1926 

30 

.8616 

1.6977 

.9924 

1.0385 

30 

.9304 

2.5386 

1.1256 

1.1956 

40 

.8631 

1.7090 

.9950 

1.0414 

40 

.9315 

2.5605 

1.1280 

1.1985 

50 

.8646 

1.7205 

.9975 

1.0443 

50 

.9325 

2.5826 

1.1304 

1.2014 

60°  00' 

.8660 

1.7321 

1.0000 

1.0472 

69°  00' 

.9336 

2.6051 

1.1328 

1.2043 

10 

.8675 

1.7437 

1.0025 

1.0501 

10 

.9346 

2.6279 

1.1352 

1.2072 

20 

.8689 

1.7556 

1.0050 

1.0530 

20 

.9356 

2.6511 

1.1376 

1.2101 

30 

.8704 

1.7675 

1.0075 

1.0559 

30 

.9367 

2.6746 

1.1400 

1.2130 

40 

.8718 

1.7796 

1.0101 

1.0588 

40 

.9377 

2.6985 

1.1424 

1.2159 

50 

.8732 

1.7917 

1.0126 

1.0617 

50 

.9387 

2.7228 

1.1448 

1.2188 

61°  00' 

.8746 

1.8040 

1.0151 

1.0647 

70°  00' 

.9397 

2.7475 

1.1472 

1.2217 

10 

.8760 

1.8165 

1.0176 

1.0676 

10 

.9407 

2.7725 

1.1495 

1.2246 

20 

.8774 

1.8291 

1.0201 

1.0705 

20 

.9417 

2.7980 

1.1519 

1.2275 

30 

.8788 

1.8418 

1.0226 

1.0734 

30 

.9426 

2.8239 

1.1543 

1.2305 

40 

.8802 

1.8546 

1.0251 

1.0763 

40 

.9436 

2.8502 

1.1567 

1.2334 

50 

.8816 

1.8676 

1.0276 

1.0^2 

.50 

.9446 

2.8770 

1.1590 

1.2363 

62°  00' 

.8829 

1.8807 

1.0301 

1.0821 

71°00' 

.9455 

2.^)042 

1.1614 

1.2392 

10 

.8843 

1.8940 

1.0326 

1.0850 

10 

.9465 

2.9319 

1.1638 

1.2421 

20 

.8857 

1.9074 

1.0351 

1.0879 

20 

.9474 

2.9600 

1.1661 

1.2450 

;io 

.8870 

1.9210 

1.0375 

1.0908 

30 

.9483 

2.9887 

1.1685 

1.2479 

40 

.8884 

1.9347 

1.0400 

1.0937 

40 

.9492 

3.0178 

1.1709 

1.2508 

50 

.8897 

1.9486 

1.0425 

1.0966 

50 

.9502 

3.0475 

1.1732 

1.2537 

63°  00' 

.8910 

1.9626 

1.0450 

1.0996 

72°  00' 

.9511 

3.0777 

1.1756 

1.2566 

I] 

Quantities  Determined  by  a  Given  Angle 

vii 

Angle 

Sine 

Tan- 
gent 

Chord 

Arc 

Angle 

Sine 

Tan- 
gent 

Chord 

Arc 

72°  00' 

.9511 

3.0777 

1.1756 

1.2566 

81°  00' 

.9877 

6.3138 

1.2989 

1.4137 

10 

.9520 

3.1084 

1.1779 

1.2595 

10 

.9881 

6.4348 

1.3011 

1.4166 

20 

.9528 

3.1397 

1.1803 

1.2625 

20 

.9886 

6.5606 

1.3033 

1.4195 

30 

.9537 

3.1716 

1.1826 

1.2654 

30 

.9890 

6.6912 

1.3055 

1.4224 

40 

.9546 

3.2041 

1.1850 

1.2683 

40 

.9894 

6.8269 

1.3077 

1.4254 

50 

.9555 

3.2371 

1.1873 

1.2712 

50 

.9899 

6.9682 

1.3099 

1.4283 

73°  00' 

.9563 

3.2709 

1.1896 

1.2741 

82°  00' 

.9903 

7.1154 

1.3121 

1.4312 

10 

.9572 

3.3052 

1.1920 

1.2770 

10 

.9907 

7.2687 

1,3143 

1.4341 

20 

.9580 

3.3402 

1.1943 

1.2799 

20- 

.9911 

7.4287 

1.3165 

1.4370 

30 

.9588 

3.3759 

1.1966 

1.2828 

30 

.9914 

7.5958 

1.3187 

1.4399 

40 

.9596 

3.4124 

1.1990 

1.2857 

40 

.9918 

7.7704 

1.3209 

1.4428 

50 

.9605 

3.4495 

1.2013 

1.2886 

50 

.9922 

7.9530 

1.3231 

1.4457 

74° 00' 

.9613 

3.4874 

1.2036 

1.2915 

83°  00' 

.9925 

8.1443 

1.3252 

1.4486 

10 

.9621 

3.5261 

1.2060 

1.2945 

10 

.9929 

8.3450 

1.3274 

1.4515 

20 

.9628 

3.5656 

1.2083 

1.2974 

20 

.9932 

8.5555 

1.3296 

1.4544 

30 

.9636 

3.6059 

1.2106 

1.3003 

30 

.9936 

8.7769 

1.3318 

1.4573 

40 

.9644 

3.6470 

1.2129 

1.3032 

40 

.9939 

9.0098 

1.3339 

1.4603 

50 

.9652 

3.6891 

1.2152 

1.3061 

50 

.9942 

9.2553 

1.3361 

1.4632 

75° 00' 

.9659 

3.7321 

1.2175 

1.3090 

84°  00' 

.9945 

9.5144 

1.3383 

1.4661 

10 

.9667 

3.7760 

1.2198 

1.3119 

10 

.9948 

9.7882 

1.3404 

1.4690 

20 

.9674 

3.8208 

1.2221 

1.3148 

20 

.9951 

10.078 

1.3426 

1.4719 

30 

MSI 

3.8667 

1.2244 

1.3177 

30 

.9954 

10.385 

1.3447 

1.4748 

40 

.9689 

3.9136 

1.2267 

1.320(5 

40 

.9957 

10.712 

1.3469 

1.4777 

50 

.9696 

3.9617 

1.2290 

1.3235 

50 

.9959 

11.059 

1.3490 

1.4806 

76° 00' 

.9703 

4.0108 

1.2313 

1.3265 

85° 00' 

.9962 

11.430 

1.3512 

1.4835 

10 

.9710 

4.0611 

1.2336 

1.3294 

10 

.9964 

11.826 

1.3533 

1.4864 

20 

.9717 

4.1126 

1.2359 

1.3323 

20 

.9967 

12.251 

1.3555 

1.4893 

30 

.9724 

4.1653 

1.2382 . 

1.3352 

30 

.9969 

12.706 

1.3576 

1.4923 

40 

.9730 

4.2193 

1.2405 

1.3381 

40 

.9971 

13.197 

l.;%97 

1.4952 

50 

.9737 

4.2747 

1.2428 

1.3410 

50 

.9974 

13.727 

1.3619 

1.4981 

77°  00' 

.9744 

4.3315 

1.2450 

1.3439 

86°  00' 

.9976 

14.301 

1.3640 

1.5010 

10 

.9750 

4.3897 

1.2473 

1.3468 

10 

.9978 

14.924 

1.3661 

1.5039 

20 

.9757 

4.4494 

1.2496 

1.3497 

20 

.9980 

15.605 

1.3682 

1.5068 

30 

.9763 

4.5107 

1.2518 

1.3526 

30 

.9981 

16.350 

1.3704 

1..5097 

40 

.9769 

4.5736 

1,2541 

1.3555 

40 

.9983 

17.169 

1.3725 

1.5126 

50 

.9775 

4.6382 

1.2564 

1.3584 

50 

.9985 

18.075 

1.3746 

1.5155 

78°  00' 

.9781 

4.7046 

1.258(i 

1.3614 

87°  00' 

.9986 

19.081 

1.3767 

1.5184 

10 

.9787 

4.7729 

1.2609 

1.3643 

10 

.9988 

20.206 

1.3788 

1.5213 

20 

.9793 

4.8430 

1.2632 

1.3672 

20 

.9989 

21.470 

1..3809 

1.5243 

30 

.9799 

4.9152 

1.2654 

1.3701 

30 

.9990 

22.904 

1.3830 

1.5272 

40 

.9805 

4.9894 

1.2677 

1.3730 

40 

.9992 

24.542 

1.3851 

1.5301 

50 

.9811 

5.0658 

1.2699 

1.3759 

50 

.9993 

26.432 

1.3872 

1.5330 

79°  00' 

.9816 

5.1446 

1.2722 

1.3788 

88°  00' 

.9994 

28.636 

1.3893 

1.5359 

10 

.9822 

5.2257 

1.2744 

1.3817 

10 

.9995 

31.242 

1.3914 

1.5388 

20 

.9827 

5.3093 

1.2766 

1.3846 

20 

.9996 

34.368 

1.3935 

1.5417 

30 

•9833 

5.3955 

1.2789 

1.3875 

30 

.9997 

38.188 

1.3956 

1.5446 

40 

.9838 

5.4845 

1.2811 

1.3904 

40 

.9997 

42.964 

1.3977 

1.5475 

50 

.9843 

5.5764 

1.2833 

1.3934 

50 

.9998 

49.104 

1.3997 

1.5504 

80°  00' 

.9848 

5.6713 

1.2856 

1.3963 

89°  00' 

.9998 

57.290 

1.4018 

1.5533 

10 

.9853 

5.7694 

1.2878 

1.3992 

10 

.9999 

68.750 

1.4039 

1.5563 

20 

.9858 

5.8708 

1.2900 

1.4021 

20 

.9999 

85.940 

1.4060 

1.5592 

30 

.9863 

5.9758 

1.2922 

1.4050 

30 

1.0000 

114.59 

1.4080 

1.5621 

40 

.9868 

6.0844 

1.2945 

1.4079 

40 

1.0000 

171.89 

1.4101 

1.5650 

50 

.9872 

6.1970 

1.2967 

1.4108 

50 

1.0000 

343.77 

1.4122 

1.5679 

81° 00' 

.9877 

6.3138 

1.2989 

1.4137 

90°  00' 

1.0000 

1.4142 

1.5708 

TABLE  II  — POWERS  AND  ROOTS 

EXPLANATION   OF   TABLE   II 

1.  Squares  and  Cubes.  The  squares  of  numbers  between 
1.00  and  10.00  at  intervals  of  .01  are  given  in  column  headed 
n^  To  find  the  square  of  any  other  number,  divide  (or  mul- 
tiply) the  given  number  by  10  to  reduce  it  to  a  number  between 
1  and  10;  find  the  square  of  this  last  number;  multiply  (or 
divide)  the  square  thus  found  by  10  twice  as  many  times  as 
you  did  the  given  number. 

The  cube  is  given  in  the  column  headed  n^.  To  find  the  cube 
of  any  number  not  between  1  and  10,  first  reduce  that  number 
to  a  number  between  1  and  10  by  dividing  (or  multiplying)  by 
a  power  of  10.  Multiply  (or  divide)  the  result  found  by  three 
times  as  high  a  power  of  10  as  was  used  to  reduce  the  given 
number. 

2.  Square  Roots.  The  square  roots  of  numbers  between  1 
and  10  are  found  in  the  column  headed  Vn. 

The  square  roots  of  numbers  between  10  and  100  may  be 
found  in  the  column  headed  VlO  n. 

The  square  roots  of  numbers  between  100  and  1000  may  be 
found  in  the  column  headed  Vn  by  multiplying  the  given  root 
by  10,  since  VlOOw  =  10  Vn. 

Other  square  roots  may  be  found  in  a  similar  manner. 

3.  Cube  Roots.     The  column  headed : 

Vn  gives  cube  roots  of  numbers  between  1  and  10 ; 
VlOn  gives  cube  roots  of  numbers  between  10  and  100 ; 
VlOOn  gives  cube  roots  of  numbers  between  100  and  1000. 

To  find  the  cube  root  of  a  number  between  1000  and  10000, 
take  10  times  the  value  found  in  the  column  headed  Vn,  since 
VlOOOn  =  10  Vn. 

Other  cube  roots  may  be  found  similarly. 


viii 


11] 

Table  II  —  Powers  and  Roots 

ix 

n 

n^ 

Vn 

VIO^ 

n^ 

</Tl 

</10n 

</WOn 

1.00 

1.0000 

1.00000 

3.16228 

1.00000 

1.00000 

2.15443 

4.64159 

1.01 
1.02 
1.03 

1.04 
1.05 
1.06 

1.07 
1.08 
1.09 

1.0201 
1.0404 
1.0609 

1.0816 
1.1025 
1.1236 

1.1449 
1.1664 

1.1881 

1.004^(9 
1.00995 
1.01489 

1.01980 
1.02470 
1.02956 

1.03441 
1.03923 
1.04403 

3.17805 
3.19374 
3.20936 

3.22490 
3.24037 
3.25576 

3.27109 
3.28634 
3.30151 

1.03030 
1.06121 
1.09273 

1.12486 
1.15762 
1.19102 

1.22504 
1.25971 
1.29503 

1.00332 
1.00662 
1.00990 

1.01316 
1.01640 
1.01961 

1.02281 
1.02599 
1.02914 

2.16159 
2.16870 
2.17577 

2.18279 
2.18976 
2.19669 

2.20358 
2.21042 
2.21722 

4.65701 
4.67233 
4.68755 

4.70267 
4.71769 
4.73262 

4.74746 
4.76220 
4.77686 

1.10 

1.2100 

1.04881 

3.31662 

1.33100 

1.03228 

2.22398 

4.79142 

1.11 
1.12 
1.13 

1.14 
1.15 
1.16 

1.17 
1.18 
1.19 

1.2321 
1.2544 
1.2769 

1.2996 
1.3225 
1.3456 

1.3689 
1.3924 
1.4161 

1.05357 
1.05830 
1.061301 

1.06771 
1.07238 
1.07703 

1.08167 
1.08628 
1.09087 

3.33167 
3.34664 
3.36155 

3.37639 
3.39116 
3.40588 

3.42053 
3.43511 
3.44964 

1.36763 
1.40493 
1.44290 

1.48154 

1.52088 
1.56090 

1.60161 
1.64303 
1.08516 

1.03540 
1.03850 
1.04158 

1.04464 
1.04769 
1.05072 

1.05373 
1.05672 
1.05970 

2.23070 
2.23738 
2.24402 

2.25062 
2.25718 
2.26370 

2.27019 
2.27664 
2.28305 

4.80590 
4.82028 
4.83459 

4.84881 
4.80294 
4.87700 

4.89097 
4.90487 
4.91868 

1.20 

1.4400 

1.09545 

3.46410 

1.72800 

1.06266 

2.28943 

4.93242 

1.21 
1.22 
1.23 

1.24 
1.25 
1.26 

1.27 
1.28 
1.29 

1.4641 
1.4884 
1.5129 

1.5376 
1.5625 
1.5876 

1.6129 
1.6384 
1.6641 

1.10000 
1.10454 
1.10905 

1.11355 
1.11803 
1.12250 

1.12694 
1.13137 
1.13578 

3.47851 
3.49285 
3.50714 

3.52136 
3.53553 
3.54965 

3.56371 
3.57771 
3.59166 

1.77156 
1.81585 
1.86087 

1.906G2 
1.95312 
2.00038 

2.04838 
2.09715 
2.14669 

1.06560 
1.06853 
1.07144 

1.07434 
1.07722 
1.08008 

1.08293 
1.08577 
1.08859 

2.29577 
2.30208 
2.30835 

2.31459 
2.32079 
2.32697 

2.33311 
2.33921 
2.34529 

4.94609 
4.95968 
4.97319 

4.98663 
5.00000 
5.01330 

5.02653 
5.03968 
5.05277 

1.30 

1.6900 

1.14018 

3.60555 

2.19700 

1.09139 

2.35133 

5.06580 

1.31 
1.32 
1.33 

1.34 
1.35 
1.36 

1.37 
1.38 
1.39 

1.7161 
1.7424 
1.7689 

1.7956 
1.8225 
1.849(> 

1.8769 
1.^)044 
1.9321 

1.14455 
1.14891 
1.15326 

1.15758 
1.16190 
1.16619 

1.17047 
1.17473 
1.17898 

3.61939 
3.63318 
3.64692 

3.66060 
3.67423 
3.68782 

3.70135 
3.71484 
3.72827 

2.24809 
2.29997 
2.35264 

2.40610 
2.46038 
2.51546 

2.57135 

2.62807 
2.68562 

1.09418 
1.09()fK) 
1.09972 

1.10247 
1.10521 
1.10793 

1.11064 
1.11334 
1.11602 

2.35735 
2.36333 
2.36928 

2.37521 
2.38110 
2.38697 

2.39280 
2.39861 
2.40439 

5.07875 
6.09164 
5.10447 

5.11723 
5.12993 
5.14256 

5.15514 
5.16765 
5.18010 

1.40 

1.9600 

1.18322 

3.74166 

2.74400 

1.11869 

2.41014 

5.19249 

1.41 
1.42 
1.43 

1.44 
1.45 
1.46 

1.47 

1.48 
1.49 

1.9881 
2.0164 
2.0449 

2.0736 
2.1025 
2.1316 

2.1609 
2.1904 
2.2201 

1.18743 
1.19164 
1.19583 

1.20000 
1.20416 
1.20830 

1.21244 
1.21655 
1.22066 

3.75500 
3.76829 
3.78153 

3.79473 
3.80789 
3.82099 

3.83406 
3.84708 
3.86005 

2.80322 
2.86329 
2.92421 

2.98598 
3.04862 
3.11214 

3.17652 
3.24179 
3.30795 

1.12135 
1.12399 
1.12662 

1.12924 
1.13185 
1.13445 

1.13703 
1.13960 
1.14216 

2.41587 
2.42156 
2.42724 

2.43288 
2.43850 
2.44409 

2.44966 
2.45520 
2.46072 

5.20483 
5.21710 
5.22932 

5.24148 
5.25359 
5.26564 

5.27763 
5.28957 
5.30146 

X 

P( 

)wers  and  Roots 

[11 

n 

n^ 

Vn 

VlOw 

1 

VlOn 

n^ 

^n 

S/lOOw 

1.50 

2.2500 

1.22474 

3.87298 

3.37500 

1.14471 

2.46621 

5.31329 

1.51 
1.52 
1.53 

1.54 
1.55 
1.56 

1.57 
1.58 
1.59 

2.2801 
2.3104 
2.3409 

2.3716 
2.4025 
2.4336 

2.4649 
2.4964 
2.5281 

1.22882 
1.23288 
1.23693 

1.24097 
1.24499 
1.24900 

1.25300 
1.25698 
1.20095 

3.88587 
3.89872 
3.91152 

3.92428 
3.93700 
3.94968 

3.96232 
3.97492 

3.98748 

3.44295 
3.51181 
3.58158 

3.65226 
3.72388 
3.79642 

3.86989 
3.94431 
4.01968 

1.14725 
1.14978 
1.15230 

1.15480 
1.15729 
1.15978 

1.16225 
1.16471 
1.16717 

2.47168 
2.47712 
2.48255 

2.48794 
2.49332 
2.49867 

2.50399 
2.509i50 
2.51458 

5.32507 
5.33680 
5.34848 

5.36011 
5.37169 
5.38321 

5.39469 
5.40612 
5.41750 

1.60 

2.5600 

1.26491 

4.00000 

4.09600 

1.169(51 

2.51984 

5.42884 

1.61 
1.62 
1.63 

1.64 
1.65 
1.66 

1.67 
1.68 
1.69 

2.5921 
2.6244 
2.6569 

2.6896 
2.7225 
2.7556 

2.7889 
2.8224 
2.8561 

1.26886 
1.27279 
1.27671 

1.28062 
1.28452 
1.28841 

1.29228 
1.29615 
1.30000 

4.01248 
4.02492 
4.03733 

4.04969 
4.06202 
4.07431 

4.08(556 
4.09878 
4.11096 

4.17328 
4.25153 
4.33075 

4.41094 
4.4i)2l2 
4.57430 

4.65746 
4.74163 
4.82681 

1.17204 
1.17446 
1.17687 

1.17927 
1.181(57 
1.18405 

1.18642 

1.18878 
1.19114 

2.52508 
2.5;?030 
2.53549 

2.54007 
2.54582 
2.55095 

2.55607 
2.56116 
2.56623 

5.44012 
5.45136 
5.46256 

5.47370 
5.48481 
5.49586 

5.50688 
5.51785 
5.52877 

1.70 

2.8900 

1.30384 

4.12311 

4.91300 

1.19348 

2.57128 

5.53966 

1.71 
1.72 
1.73 

1.74 
1.75 
1.76 

1.77 

1.78 
1.79 

2.9241 
2.9584 
2.9929 

3.0276 
3.0625 
3.0976 

3.1329 
3.1684 
3.2041 

1.30767 
1.31149 
1.31529 

1.31909 
1.32288 
1.32665 

1.33041 
1.33417 
1.33791 

4.13521 
4.14729 
4.15933 

4.17133 
4.18330 
4.19524 

4.20714 
4.21900 
4.23084 

5.00021 
5.08845 
5.17772 

5.26802 
5.35938 
5.45178 

5.54523 
5.63975 
5.73534 

1.19582 
1.19815 
1.20046 

1.20277 
1.20507 
1.20736 

1.20964 
1.21192 
1.21418 

2.57631 
2.58133 
2.58632 

2.59129 
2.59625 
2.60118 

2.60610 
2.61100 
2.61588 

5.55050 
5.56130 
5.57205 

5.58277 
5.59344 
5.60408 

5.61467 
5.62523 
5.63574 

1.80 

3.2400 

1.34164 

4.24264 

5.83200 

1.21644 

2.62074 

5.64622 

1.81 
1.82 
1.83 

1.84 
1.85 
1.86 

1.87 
1.88 
1.89 

3.2761 
3.3124 
3.3489 

3.3856 
3.4225 
3.4596 

3.4969 
3.5344 
3.5721 

1.34536 
1.34907 
1.35277 

1.35647 
1.36015 
1.36382 

1.36748 
1.37113 
1.37477 

4.25441 
4.26615 
4.27785 

4.28952 
4.30116 
4.31277 

4.32435 
4.3351K) 
4.34741 

5.92974 
6.02857 
6.12849 

6.22950 
6.33162 
6.43486 

6.53920 
6.64467 
6.75127 

1.21869 
1.22093 
1.22316 

1.22539 
1.227(50 
1.22981 

1.23201 
1.23420 
1.23639 

2.(52559 
2.63041 
2.63522 

2.64001 
2.(54479 
2.64954 

2.65428 
2.65901 
2.66371 

5.65665 
5.66705 
5.67741 

5.68VV3 
5.69802 
5.70827 

5.71848 
5.72865 
5.73879 

1.90 

3.6100 

1.37840 

4.35890 

6.85900 

1.23856 

2.66840 

5.74890 

1.91 
1.92 
1.93 

1.94 
1.95 
1.96 

1.97 
1.98 
1.99 

3.6481 
3.6864 
3.7249 

3.7636 
3.8025 
3.8416 

3.8809 
3.9204 
3.9601 

1.38203 
1.38564 
1.38924 

1.39284 
1.39642 
1.40000 

1.40357 
1.40712 
1.41067 

4.37035 
4.38178 
4.39318 

4.40454 
4.41588 
4.42719 

4.43847 
4.44972 
4.46094 

6.96787 
7.07789 
7.1890(J 

7.30138 
7.41488 
7.52954 

7.64537 
7.76239 
7.880(50 

1.24073 
1.24289 
1.24.505 

1.24719 
1.24933 
1.25146 

1.25a59 
1.25571 
1.25782 

2.67307 
2.67773 
2.68237 

2.68700 
2.69161 
2.69620 

2.70078 
2.70534 
2.70989 

5.75897 
5.76900 
5.77900 

5.78896 
5.79889 
5.80879 

5.81865 
5.82848 
5.83827 

H] 

Powers  and  Roots 

xi 

n 

n^ 

v^ 

VlOn 

n^ 

^ 

<^10n 

■ 

</100n 

2.00 

4.0000 

1.41421 

4.47214 

8.00000 

1.25992 

2.71442 

5.84804 

2.01 
2.02 
2.03 

2.04 
2.05 
2.06 

2.07 
2.08 
2.09 

4.0401 
4.0804 
4.1209 

4.1616 
4.2025 
4.2436 

4.2849 
4.3264 
4.3681 

1.41774 
1.42127 
1.42478 

1.42829 
1.43178 
1.43527 

1.43875 
1.44222 
1.44568 

4  48330 
4.4<>i44 
4.50555 

4.51664 
4.52769 
4.53872 

4.54973 
4.56070 
4.57165 

8.120(^0 
8.24241 
8.36543 

8.48966 
8.61512 
8.74182 

8.86974 
8.99891 
9.12933 

1.26202 
1.26411 
1.26619 

1.2(5827 
1.27033 
1.27240 

1.27445 
1.27650 
1.27854 

2.71893 
2.72344 
2.72792 

2.73239 
2.73(585 
2.74129 

2.74572 
2.75014 
2.75454 

5.85777 
5.86746 
5.87713 

5.88677 
5.89637 
5.90594 

5.91548 
5.92499 
5.93447 

2.10 

4.4100 

1.44914 

4.58258 

9  26100 

1.28058 

2.75892 

5.94392 

2.11 
2.12 
2.13 

2.14 
2.15 
2.16 

2.17 
2.18 
2.19 

4.4521 
4.4944 
4.5369 

4.5796 
4.6225 
4.6656 

4.7089 
4.7524 
4.7961 

1.45258 
1.45602 
1.45945 

1.46287 
1.46629 
1.46969 

1.47309 
1.47648 

1.47986 

4.59347 
4.60435 
4.61519 

4.62601 
4.63681 
4.64758 

4.65833 
4.66905 
4.67974 

9.39393 
9.52813 
9.663(50 

9.80034 
9.93838 
10.0777 

10.2183 
10.3602 
10.5035 

1.28261 
1.28463 
1.28665 

1.28866 
1.2fX)66 
1.29266 

1.29465 
1.29664 
1.29862 

2.76330 
2.76766 
2.77200 

2.77633 

2.78065 
2.78495 

2.78924 
2.79352 
2.79779 

5.95334 
5.96273 
5.97209 

5.98142 
5.99073 
6.00000 

6.00925 
6.01846 
6.02765 

2.20 

4.8400 

1.48324 

4.69042 

10.6480 

1.30059 

2.80204 

6.03(581 

2.21 
2.22 
2.23 

2.24 
2.25 
2.26 

2.27 
2.28 
2.29 

4.8841 
4.9284 
4.9729 

5.0176 
5.0625 
5.1076 

5.1529 
5.1984 
5.2441 

1.486(51 
1.48997 
1.49332 

1.49666 
1.50000 
1.50333 

1.50665 
1.50997 
1.51327 

4.70106 
4.71169 
4.72229 

4.73286 
4.74342 
4.75395 

4.76445 
4.77493 
4.78539 

10.7939 
10.9410 
11.0896 

11.2394 
11.3906 
11.5432 

11.6971 
11.8524 
12.00<)0 

1.302.56 
1.30452 
1.30648 

1.30843 
1.31037 
1.31231 

1.31424 
1.31617 
1.31809 

2.80628 
2.81050 
2.81472 

2.81892 
2.82311 
2.82728 

2.83145 
2.83560 
2.83974 

6.04594 
6.05505 
6.06413 

6.07318 
6.08220 
6.09120 

6.10017 
6.10911 
6.11803 

2.30 

5.2{)00 

1.51658 

4.79583 

12.1670 

1.32001 

2.84387 

6.12693 

2.31 
2.32 
2.33 

2.34 
2.35 
2.36 

2.37 
2.38 
2.39 

5.3361 
5.3824 
5.4289 

5.4756 
5.5225 
5.5696 

5.6169 
5.6644 
5.7121 

1.51987 
1.52315 
1.52643 

1.52971 
1.53297 
1.53623 

1.53948 
1.54272 
1.54596 

4.80(525 
4.81664 
4.82701 

4.83735 
4.84768 
4.85798 

4.86826 
4.87852 
4.88876 

12.3264 
12.4872 
12.6493 

12.8129 
12.9779 
13.1443 

13.3121 
13.4813 
13.6519 

1.32192 
1.32382 
1.32572 

1.32761 
1.32950 
1.33139 

1.33326 
1.33514 
1.33700 

2.84798 
2.85209 
2.85618 

2.86026 
2.86433 
2.86838 

2.87243 
2.87646 
2.88049 

6.13579 
6.14463 
6.15345 

6.16224 
6.17101 
6.17975 

6.18846 
6.19715 
6.20582 

2.40 

5.7600 

1.54919 

4.89898 

13.8240 

1.33887 

2.88450 

6.21447 

2.41 
2.42 
2.43 

2.44 
2.45 
2.46 

2.47 
2.48 
2.49 

5.8081 
5.8564 
5.9049 

5.9536 
6.0025 
6.0516 

6.1009 
6.1504 
6.2001 

1.55242 
1.55563 
1.55885 

1.56205 
1.56525 
1.56844 

1.57162 
1.57480 
1.57797 

4.90918 
4.91935 
4.92950 

4.93964 
4.^)4975 
4.95984 

4.96991 
4.97996 
4.98999 

13.9975 
14.1725 
14.3489 

14.5268 
14.7061 
14.8869 

15.0692 
15.2530 
15.4382 

1.M072 
1.34257 
1.34442 

1.34626 
1.34810 
1.34993 

1.35176 
1.35358 
1.35540 

2.88850 
2.89249 
2.89647 

2.90044 
2.90439 
2.90834 

2.91227 
2.91620 
2.92011 

6.22308 
6.23168 
6.24025 

6.24880 
6.25732 
6.26583 

6.27431 
6.28276 
6.29119 

xii 

Powers  and  Roots 

Cn 

n 

n^ 

v^ 

^/10n 

n^ 

^n 

^10  n 

s/lOOn 

2.50 

6.2500 

1.58114 

5.00000 

15.6250 

1.35721 

2.92402 

6.29961 

2.51 
2.52 
2.53 

2.54 
2.55 
2.56 

2.57 
2.58 
2.59 

6.3001 
6.3504 
6.4009 

6.4516 
6.5025 
6.5536 

6.6049 
6.6564 
6.7081 

1.58430 

1.58745 
1.59060 

1.59374 
1.59687 
1.60000 

1.60312 
1.60624 
1.60935 

5.00999 
5.01996 
5.02991 

5.03984 
5.04975 
5.05964 

5.06952 
5.07937 
5.08920 

15.8133 
16.0030 
16.1943 

16.3871 
16.5814 
16.7772 

16.9746 
17.1735 
17.3740 

1.35902 
1.36082 
1.30202 

1.36441 
1.36020 
1.30798 

1.30970 
1.37153 
1.37330 

2.92791 
2.93179 
2.93507 

2.93953 
2.94338 
2.94723 

2.95106 

2.95488 
2.95869 

6.30799 
6.31636 
6.32470 

6.33303 
6.34133 
6.34960 

6.35786 
6.36610 
6.37431 

2.60 

6.7600 

1.61245 

5.09902 

17.5700 

1.. 37507 

2.96250 

6.38250 

2.61 
2.62 
2.63 

2.64 
2.65 
2.66 

2.67 
2.68 
2.69 

6.8121 
6.8644 
6.9169 

6.9696 
7.0225 
7.0756 

7.1289 
7.1824 
7.2361 

1.61555 
1.61864 
1.62173 

1.62481 

1.62788 
1.63095 

1.63401 
1.03707 
1.04012 

5.10882 
5.11859 
5.12835 

5.13809 
5.14782 
5.15752 

5.16720 

5.17687 
5.18052 

17.7796 
17.9847 
18.1914 

18.3997 
18.0096 
18.8211 

19.0342 
19.2488 
19.4051 

1.37083 
1.37859 
1.38034 

1.38208 
1.38383 
1.38557 

1.38730 
1.38903 
1.39070 

2.90029 
2.97007 
2.97385 

2.97761 
2.98137 
2.98511 

2.98885 
2.99257 
2.99629 

6.39068 
6.39883 
6.40696 

6.41507 
0.42316 
6.43123 

6.43928 
6.44731 
6.45531 

2.70 

7.2900 

1.64317 

5.19f)15 

19.0830 

1.39248 

3.00000 

6.46330 

2.71 

2.72 
2.73 

2.74 
2.75 
2.76 

2.77 
2.78 
2.79 

7.3441 
7.3984 
7.4529 

7.5076 
7.5625 
7.6176 

7.6729 

7.7284 
7.7841 

1.64621 
1.64924 
1.65227 

1.65529 
1.65831 
1.66132 

1.66433 
1.60733 
1.07033 

5.20577 
5.21536 
5.22494 

5.23450 
5.24404 
5.25357 

5.26308 
5.27257 
5.28205 

19.9025 
20.1236 
20.3464 

20.5708 
20.7969 
21.0246 

21.2539 
21.4850 
21.7176 

1.39419 
1.39591 
1.39701 

1.39932 
1.40102 
1.40272 

1.40441 
1.40610 
1.40778 

3.00370 
3.00739 
3.01107 

3.01474 
3.01841 
3.02206 

3.02570 
3.02934 
3.03297 

6.47127 
6.47922 
6.48715 

6.49507 
6.5029<j 
6.51083 

6.51868 
6.52()52 
6.53434 

2.80 

7.8400 

1.07332 

5.29150 

21.9520 

1.40946 

3.03659 

6.54213 

2.81 
2.82 
2.83 

2.84 
2.85 
2.86 

2.87 
2.88 
2.89 

7.8901 
7.9524 
8.0089 

8.0656 
8.1225 
8.1796 

8.2369 
8.2944 
8.3521 

1.67631 
1.67929 
1.68226 

1.68523 
1.68819 
1.69115 

1.69411 
1.69706 
1.70000 

5.30094 
5.31037 
5.31977 

5.32917 
5.33854 
5.34790 

5.35724 
5.36656 
5.37587 

22.1880 
22.4258 
22.6052 

22.9063 
23.1491 
23.3937 

23.6399 
23.8879 
24.1376 

1.41114 
1.41281 
1.41448 

1.41614 
1.41780 
1.41946 

1.42111 
1.42276 
1.42440 

3.04020 
3.04380 
3.04740 

3.05098 
3.05456 
3.05813 

3.06169 
3.06524 
3.0()878 

6.54991 
6.55767 
6.56541 

6.57314 
6.58084 
6.58853 

6.59620 
6.60385 
6.61149 

2.90 

8.4100 

1.70294 

5.38516 

24.3890 

1.42004 

3.07232 

6.61911 

2.91 
2.92 
2.93 

2.94 
2.95 
2.96 

2.97 
2.98 
2.99 

8.4681 
8.5264 
8.5849 

8.6436 
8.7025 
8.7616 

8.8209 
8.8804 
8.9401 

1.70587 
1.70880 
1.71172 

1.71464 
1.71756 
1.72047 

1.72337 
1.72627 
1.72916 

5.39444 
5.40370 
5.41295 

6.42218 
5143139 
5.44059 

5.44977 
5.45894 
5.46809 

24.6422 
24.8971 
25.1538 

25.4122 
25.6724 
25.9343 

26.1981 
26.4036 
26.7309 

1.42708 
1.42931 
1.430M 

1.43257 
1.43419 
1.43581 

1.43743 
1.43904 
1.44005 

3.07584 
3.07936 
3.08287 

3.086.38 
3.08987 
3.09336 

3.09084 
3.10031 
3.10378 

6.62671 
6.63429 
6.64185 

6.64940 
6.65693 
6.66444 

6.67194 
6.07942 
6.68688 

11] 


Powers  and  Boots 


Xlll 


n 

n^ 

V^ 

VlOn 

n^ 

^ 

^10** 

^100  n 

3.00 

9.0000 

1.73205 

5.47723 

27.0000 

1.44225 

3.10723 

6.69433 

3.01 
3.02 
3.03 

3.04 
3.05 
3.06 

3.07 
3.08 
3.09 

9.0601 
9.1204 
9.1809 

9.2416 
9.3025 
9.3636 

9.4249 
9.4864 
9.5481 

1.73494 
1.73781 
1.74069 

1.74356 
1.74642 
1.74929 

1.75214 
1.75499 
1.75784 

5.48635 
5.49545 
5.50454 

5.51362 
5.52268 
5.53173 

5.54076 
5.54977 
5.55878 

27.2709 
27.5436 
27.8181 

28.0945 
28.3726 
28.6526 

28.9344 
29.2181 
29.5036 

1.44385 
1.44545 
1.44704 

1.44863 
1.45022 
1.45180 

1.45338 
1.45496 
1.45653 

3.11068 
3.11412 
3.11756 

3.12098 
3.12440 
3.12781 

3.13121 
3.13461 
3.13800 

6.70176 

6.70917 
6.71657 

6.72395 
6.73132 
6.73866 

6.74600 
6.75331 
6.76061 

3.10 

9.6100 

1.76068 

5.56776 

29.7910 

1.45810 

3.14138 

6.76790 

3.11 
3.12 
3.13 

3.14 
3.15 
3.16 

3.17 
3.18 
3.19 

9.6721 
9.7344 
9.7969 

9.8596 
9.9225 
9.9856 

10.0489 
10.1124 
10.1761 

1.76352 
1.76635 
1.76918 

1.77200 
1.77482 
1.77764 

1.78045 
1.78326 
1.78606 

5.57674 
5.58570 
5.59464 

5.60357 
5.61249 
5.62139 

5.63028 
5.63915 
5.64801 

30.0802 
30.3713 
30.6643 

30.9591 
31.2559 
31.5545 

31.8550 
32.1574 
32.4618 

1.45967 
1.46123 
1.46279 

1.46434 
1.46590 
1.46745 

1.46899 
1.47054 
1.47208 

3.14475 
3.14812 
3.15148 

3.15483 
3.15818 
3.16152 

3.16485 
3.16817 
3.17149 

6.77517 
6.78242 
6.78966 

6.79688 
6.80409 
6.81128 

6.81846 
6.82562 
6.83277 

3.20 

10.2400 

1.78885 

5.65685 

32.7680 

1.47361 

3.17480 

6.83990 

3.21 
3.22 
3.23 

3.24 
3.25 
3.26 

3.27 
3.28 
3.29 

10.3041 
10.3684 
10.4329 

10.4976 
10.5625 
10.6276 

10.6929 
10.7584 
10.8241 

1.79165 
1.79444 
1.79722 

1.80000 
1.80278 
1.80555 

1.80831 
1.81108 
1.81384 

5.66569 
5.67450 
5.68331 

5.69210 
5.70088 
5.70964 

5.71839 
5.72713 
5.73585 

33.0762 
33.3862 
33.6983 

34.0122 
34.3281 
34.64(50 

34.9658 
35.2876 
a5.6113 

1.47515 
1.47668 
1.47820 

1.47973 
1.48125 

1.48277 

1.48428 
1.48579 
1.48730 

3.17811 
3.18140 
3.18469 

3.18798 
3.19125 
3.19452 

3.19778 
3.20104 
3.20429 

6.84702 
6.85412 
6.86121 

6.86829 
6.87534 
6.88239 

6.88942 
6.89643 
6.90344 

3.30 

3.31 
3.32 
3.33 

3.34 
3.35 
3.36 

3.37 
3.38 
3.39 

10.81X)0 

1.81659 

5.74456 

35.9370 

1.48881 

3.20753 

6.91042 

10.9561 
11.0224 
11.0889 

11.1556 
11.2225 
11.2896 

11.3569 
11.4244 
11.4921 

1.81934 
1.82209 
1.82483 

1.82757 
1.83030 
1.83303 

1.83576 
1.83848 
1.84120 

5.75326 
5.76194 
5.77062 

5.77927 
5.78792 
5.79655 

5.80517 
5.81378 
6.82237 

36.2647 
36.5944 
36.9260 

37.2597 
37.5954 
37.9331 

38.2728 
38.6145 
38.9582 

1.49181 
1.49330 

1.49480 
1.49()20 
1.49777 

1.49926 
1.50074 
1.50222 

3.21077 
3.21400 
3.21722 

3.22044 
3.22365 
3.22686 

3.2.3006 
3.23325 
3.23643 

6.91740 
6.92436 
6.93130 

6.93823 
6.94515 
6.95205 

6.95894 
6.96582 
6.97268 

3.40 

11.5600 

1.84391 

5.83095 

39.3040 

1.50369 

3.23961 

6.97953 

3.41 
3.42 
3.43 

3.44 
3.45 
3.46 

3.47 
3.48 
3.49 

11.6281 
11.6964 
11.7649 

11.8336 
11.9025 
11.9716 

12.0409 
12.1104 
12.1801 

1.84662 
1.84932 
1.85203 

1.85472 
1.85742 
1.86011 

1.86279 
1.86548 
1.86815 

5.83952 
5.84808 
5.85662 

5.86515 
5.87367 
5.88218 

5.89067 
5.89915 

5.907()2 

39.6518 
40.0017 
40.3536 

40.7076 
41.06:^ 
41.4217 

41.7819 
42.1442 
42.5085 

1.50517 
1.50664 
1.50810 

1.50957 
1.51103 
1.51249 

1.51394 
1.51540 
1.51685 

3.24278 
3.24595 
3.24911 

3.25227 
3.25542 
3.25856 

3.26169 
3.26482 
3.26795 

6.98637 
6.99319 
7.00000 

7.00680 
7.01358 
7.02035 

7.02711 
7.03385 
7.04058 

xiv 

Powers  and  Roots 

Ui 

n 

n2 

Vn 

VlOw 

W,8 

^ 

^10  It 

^100  w 

8.50 

12.2500 

1.87083 

5.91008 

42.8750 

1.51829 

3.27107 

7.04730 

3.51 
3.52 
3.53 

3.54 
3.55 
3.56 

3.57 
3.58 
3.59 

12.3201 
12.3904 
12.4009 

12.5316 
12.6025 
12.6736 

12.7449 
12.8164 

12.8881 

1.87350 
1.87617 
1.87883 

1.88149 
1.88414 
1.88680 

1.88944 
1.89209 
1.89473 

5.92453 
5.93296 
5.94138 

6.94979 
5.95819 
5.96657 

5.97495 
5.98331 
5.99166 

43.2436 
43.6142 
43.9870 

44.3619 
44.7389 
45.1180 

45.4993 

45.8827 
46.2683 

1.51974 
1.52118 
1.52262 

1.52406 
1.52549 
1.52692 

1.52835 
1.52978 
1.53120 

3.27418 
3.27729 
3.28039 

3.28348 
3.28G57 
3.28965 

3.29273 
3.29580 
3.29887 

7.05400 
7.06070 
7.06738 

7.07404 
7.08070 
7.08734 

7.09397 
7.10059 
7.10719 

3.60 

12.9600 

1.89737 

6.00000 

46.6560 

1.53202 

3.30193 

7.11379 

3.61 
3.62 
3.63 

3.64 
3.65 
3.66 

3.67 
3.68 
3.69 

13.0321 
13.1044 
13.1769 

13.2496 
13.3225 
13.3956 

13.4689 
13.5424 
13.6161 

1.90000 
1.90203 
1.90526 

1.90788 
1.91050 
1.91311 

1.91572 
1.91833 
1.92094 

6.00833 
6.01G64 
6.02495 

6.03324 
6.04152 
6.04979 

6.05805 
6.06G30 
6.07454 

47.0459 
47.4379 
47.8321 

48.2285 
48.6271 
49.0279 

49.4309 
49.8300 
50.2434 

1.53404 
1.53545 
1.53686 

1.53827 
1.53968 
1.54109 

1.54249 
1.54389 
1.. 54529 

3.30498 
3.30803 
3.31107 

3.31411 
3.31714 
3.32017 

3.32319 
3.32621 
3.32922 

7.12037 
7.12694 
7.13349 

7.14004 
7.14657 
7.15309 

7.15960 
7.16610 

7.17258 

3.70 

13.6900 

1.923.54 

6.08276 

50.6530 

1.54668 

3.33222 

7.17905 

3.71 
3.72 
3.73 

3.74 
3.75 
3.76 

3.77 
3.78 
3.79 

13.7641 
13.8384 
13.9129 

13.9876 
14.0625 
14.1376 

14.2129 
14.2884 
14..3641 

1.92014 
1.92873 
1.93132 

1.93391 
1.93649 
1.93907 

1.94165 
1.94422 
1.94679 

6.09098 
6.09918 
6.10737 

6.11555 
6.12372 
6.13188 

6.14003 
6.14817 
6.15630 

51.0648 
51.4788 
51.8951 

52.3136 
52.7344 
53.1574 

53.5826 
54.0102 
54.4399 

1.54807 
1.54946 
1.55085 

1.55223 
1.55362 
1.55500 

1.55637 
1.55775 
1.55912 

3.33522 
3.33822 
3.34120 

3.34419 
3.34716 
3.35014 

3.35310 
3.35607 
3.35902 

7.18552 
7.19197 
7.19840 

7.20483 
7.21125 
7.21765 

7.22405 
7.23043 
7.2.'i680 

3.80 

14.4400 

1.94936 

6.16441 

54.8720 

1.56049 

3.36198 

7.24316 

3.81 
3.82 
3.83 

3.84 
3.85 
3.86 

3.87 
3.88 
3.89 

14.5101 
14.5924 
14.6689 

14.745C 
14.8225 
14.8996 

14.9769 
15.0544 
15.1321 

1.95192 
1.95448 
1.95704 

1.95959 
1.96214 
1.96469 

1.96723 
1.96977 
1.97231 

6.17252 
6.180G1 
6.18870 

6.19()77 
6.20484 
6.21289 

6.22093 
6.22896 
6.23699 

55.3063 
55.7430 
56.1819 

56.6231 
57.0666 
57.5125 

57.9606 
58.4111 
58.8639 

1.56186 
1.56322 
1.56459 

1.56595 
1.56731 
1.56866 

1.57001 
1.57137 
1.57271 

3.36492 
3.36786 
3.37080 

3.37373 
3.37666 
3.37958 

3.38249 
3.38540 
3.38831 

7.24950 
7.25584 
7.26217 

7.26848 
7.27479 
7.28108 

7.28736 
7.293(53 
7.29989 

3.90 

15.2100 

1.97484 

6.24500 

59.3190 

1.57406 

3.39121 

7.30614 

3.91 
3.92 
3.93 

3.94 
3.95 
3.96 

3.97 
3.98 
3.99 

15.2881 
15.3664 
15.4449 

15.5236 
15.6025 
15.6816 

15.7009 
15.8404 
15.9201 

1.97737 
1.97990 
1.98242 

1.98494 
1.98746 
1.98997 

1.99249 
1.99499 
1.99750 

6.25300 
6.26099 
6.26897 

6.27694 
6.28490 
6.29285 

6.30079 
6.30872 
6.31664 

59.7765 
60.2363 
(50.6985 

61.1630 
61.6299 
62.0991 

62.5708 
63.0448 
63.5212 

1.57541 
1.57675 
1.57809 

1.57942 
1.58076 
1.58209 

1.58342 
1.58475 
1.58<i08 

3.39411 
3.39700 
3.39988 

3.40277 
3.405<)4 
3.40851 

3.41138 
3.41424 
3.41710 

7.31238 
7.318(51 
7.32483 

7.33104 
7.33723 
7.34342 

7.34960 
7.35576 
7.36192 

II] 

Powers  and  Roots 

XV 

n 

W2 

va 

VlOn 

n3 

^ 

<^10n 

</imn 

4.00 

16.0000 

2.00000 

6.32456 

64.0000 

1.58740 

3.41905 

7.36806 

4.01 
4.02 
4.03 

4.04 
4.05 
4.06 

4.07 
4.08 
4.09 

16.0801 
16.1604 
16.2409 

16.3216 
16.4025 
16.4836 

16.5649 
16.6464 
16.7281 

2.00250 
2.00499 
2.00749 

2.00998 
2.01246 
2.01494 

2.01742 
2.01990 
2.02237 

6.33246 
6.34035 
6.34823 

6.35610 
6.3G396 
6.37181 

6.37966 
6.38749 
6.39531 

64.4812 
64.9048 
65.4508 

65.9393 
66.4301 
66.9234 

67.4191 
67.9173 
68.4179 

1.58872 
1.5fX)04 
1.59136 

1.59267 
1.59399 
1.59530 

1.59661 
1.59791 
1.59922 

3.42280 
3.42564 
3.42848 

3.43131 
3.43414 

3.43697 

3.43979 
3.44260 
3.44541 

7.37420 
7.38032 
7.38644 

7.39254 
7.39864 
7.40472 

7.41080 
7.41686 
7.42291 

4.10 

16.8100 

2.02485 

6.40312 

68.9210 

1.60052 

3.44822 

7.42896 

4.11 
4.12 
4.13 

4.14 
4.15 
4.16 

4.17 
4.18 
4.19 

16.8921 
16.9744 
17.0569 

17.1396 
17.2225 
17.3056 

17.3889 
17.4724 
17.5561 

2.02731 
2.02978 
2.03224 

2.03470 
2.03715 
2.03961 

2.04206 
2.04450 
2.04695 

6.41093 
6.41872 
6.42651 

6.43428 
6.44205 
0.44981 

6.45755 
6.46529 
6.47302 

69.4265 
69.9345 
70.4450 

70.9579 
71.4734 
71.9913 

72.5117 
73.0346 
73.5601 

1.60182 
1.60312 
1.60441 

1.60571 
1.60700 
1.60829 

1.60958 
1.61086 
1.61215 

3.45102 
3.45382 
3.45661 

3.45939 
3.46218 
3.46496 

3.46773 
3.47050 
3.47327 

7.43499 
7.44102 
7.44703 

7.45304 
7.45904 
7.46502 

7.47100 
7.47697 
7.48292 

4.20 

17.6400 

2.04939 

6.48074 

74.0880 

1.61343 

3.47603 

7.48887 

4.21 
4.22 
4.23 

4.24 
4.25 
4.26 

4.27 
4.28 
4.29 

17.7241 

17.8084 
17.8929 

17.9776 
18.0625 
18.1476 

18.2329 
18.3184 
18.4041 

2.05183 
2.05426 
2.05670 

2.05913 
2.06155 
2.06398 

2.06640 
2.0(>882 
2.07123 

6.48845 
6.49(515 
6.50384 

6.51153 
6.51920 
6.52687 

6.53452 
6.54217 
6.54981 

74.6185 
75.1514 
75.6870 

76.2250 
76.7656 
77.3088 

77.8545 

78.4028 
78.9536 

1.61471 
1.61599 
1.61726 

1.61853 
1.61981 
1.62108 

1.62234 
1.62361 
1.62487 

3.47878 
3.48154 
3.48428 

3.48703 
3.48977 
3.49250 

3.49523 
3.49796 
3.50068 

7.49481 
7.50074 
7.50666 

7.51257 
7.51847 
7.52437 

7.53025 
7.53612 
7.54199 

4.30 

18.4900 

2.07364 

6.55744 

79.5070 

1.62613 

3.50340 

7.54784 

4.31 
4.32 
4.33 

4.34 
4.35 
4.36 

4.37 
4.38 
4.39 

18.5 
18.f 
18.7 

18.8 
18.fl 
19.C 

19.C 
19.1 
19.2 

761 
«24 
489 

356 
225 

K)96 

969 
844 
721 

2.07605 

2.07846 
2.08087 

2.08327 
2.08567 
2.08806 

2.09045 
2.09284 
2.09523 

6.56506 
6.57267 
6.58027 

6.58787 
6.59545 
6.60303 

6.61060 
6.61816 
6.62571 

80.0630 
80.6216 
81.1827 

81.7465 
82.3129 
82.8819 

83.4535 

84.0277 
84.6045 

1.62739 
1.62865 
1.62991 

1.63116 
1.63241 
1.63366 

1.63491 
1.63619 
1.63740 

3.50611 
3.50882 
3.51153 

3.51423 

3.51692 
3.519G2 

3.52231 
3.52499 
3.52767 

7.55369 
7.55953 
7.56535 

7.57117 
7.57698 
7.68279 

7.58858 
7.59436 
7.60014 

4.40 

19.3600 

2.09762 

6.63325 

85.1840 

1.63864 

3.53035 

7.60590 

4.41 
4.42 
4.43 

4.44 
4.45 
4.46 

4.47 
4.48 
4.49 

19.4481 
19.5364 
19.6249 

19.7136 
19.8025 
19.8916 

19.9809 
20.0704 
20.1601 

2.10000 
2.10238 
2.10476 

2.10713 
2.10950 
2.11187 

2.11424 
2.11660 
2.11896 

6.64078 
6.64831 
6.65582 

6,66333 
6.67083 
6.67832 

6.68581 
6.69328 
6.70075 

85.7661 
86.3509 
86.9383 

87.5284 
88.1211 
88.7165 

89.3146 
89.9154 
90.5188 

1.63988 
1.64112 
1.64236 

1.64.359 
l.(i4483 
1.64606 

1.64729 
1.64851 
1.64974 

3.53302 
3.53569 
3.53835 

3.54101 
3.54367 
3.54632 

3.54897 
3.55162 
3.55426 

7.61166 
7.61741 
7.62315 

7.62888 
7.63461 
7.64032 

7.64603 
7.65172 
7.65741 

xvi 

Powers  and  Roots 

[11 

n 

W-2 

V^ 

VIO** 

n^ 

■^n 

^IQn 

S/lOOw 

4.50 

20.2500 

2.12132 

6.70820 

91.1250 

1.65096 

3.55689 

7.66309 

4.51 
4.52 
4.53 

4.54 
4.55 
4.56 

4.57 
4.58 
4.59 

20.3401 
20.4304 
20.5209 

20.6116 
20.7025 
20.7936 

20.8849 
20.9764 
21.0681 

2.12368 
2.12603 
2.12838 

2.13073 
2.13307 
2.13542 

2.13776 
2.14009 
2.14243 

6.71565 
6.72309 
6.73053 

6.73795 
6.74537 
6.75278 

6.76018 
6.76757 
6.77495 

91.7339 
92.3454 
92.9597 

93.5767 
94.1964 
94.8188 

95.4440 
96.0719 
96.7026 

1.65219 
1.65341 
1.65462 

1.65584 
1.65706 
1.65827 

1.65948 
1.66069 
1.66190 

3.55953 
3.56215 
3.56478 

3.56740 
3.57002 
3.57263 

3.57524 

3.57785 
3.58045 

7.66877 
7.67443 
7.68009 

7.68573 
7.69137 
7.69700 

7.70262 

7.70824 
7.71384 

4.60 

21.1600 

2.14476 

6.78233 

97.3360 

1.66310 

3.58305 

7.71944 

4.61 
4.62 
4.63 

4.64 
4.65 
4.66 

4.67 
4.68 
4.69 

21.2521 
21.3444 
21.4369 

21.5296 
21.6225 
21.7156 

21.8089 
21.9024 
21.9961 

2.14709 
2.14942 
2.15174 

2.15407 
2.15639 
2.15870 

2.16102 
2.16333 
2.16564 

6.78970 
6.79706 
6.80441 

6.81175 
6.81909 
6.82642 

6.83374 
6.84105 
6.84836 

97.9722 
98.6111 
99.2528 

99.8973 
100.545 
101.195 

101.848 
102.503 
103.162 

1.66431 
1.66551 
1.66671 

1.66791 
1.66911 
1.67030 

1.67150 
1.67269 
1.67388 

3.58564 
3.58823 
3.59082 

3.59340 
3.59598 
3.59856 

3.60113 
3.60370 
3.60626 

7.72503 
7.73061 
7.73619 

7.74175 
7.74731 

7.75286 

7.75840 
7.76394 
7.76946 

4.70 

22.0900 

2.16795 

6.85565 

103.823 

1.67507 

3.60883 

7.77498 

4.71 
4.72 
4.73 

4.74 
4.75 
4.76 

4.77 
4.78 
4.79 

22.1841 
22.2784 
22.3729 

22.4676 
22.5625 
22.6576 

22.7529 

22.8484 
22.9441 

2.17025 
2.17256 
2.17486 

2.17715 
2.17945 
2.18174 

2.18403 

2.18632 
2.18861 

6.86294 
6.87023 
6.87750 

6.88477 
6.89202 
6.89928 

6.90652 
6.91375 
6.92098 

104.487 
105.154 
105.824 

106.496 
107.172 
107.850 

108.531 
109.215 
109.<)02 

1.67626 
1.67744 
1.67863 

1.67981 
1.68099 
1.68217 

1.68334 
1.68452 
1.68569 

3.61138 
3.61394 
3.61649 

3.61903 
3.62158 
3.62412 

3.6?665 
3.62919 
3.63172 

7.78049 
7.78599 
7.79149 

7.79697 
7.80245 
7.80793 

7.81339 
7.81885 
7.82429 

4.80 

23.0400 

2.19089 

6.92820 

110.592 

1.68687 

3.63424 

7.82974 

4.81 

4.82 
4.83 

4.84 
4.85 
4.86 

4.87 
4.88 
4.89 

23.1361 
23.2324 
23.3289 

23.4256 
23.5225 
23.6196 

23.7169 
23.8144 
23.9121 

2.19317 
2.19545 
2.19773 

2.20000 
2.20227 
2.20454 

2.20681 
2.20907 
2.21133 

6.93542 
6.94262 
6.94982 

6.95701 
6.96419 
6.97137 

6.97854 
6.98570 
6.99285 

111.285 
111.980 
112.679 

113..380 
114.084 
114.791 

115.501 
116.214 
116.930 

1.68804 
1.68920 
1.69037 

1.69154 
1.69270 
1.69386 

1.69503 
1.69619 
1.69734 

3.63676 
3.63928 
3.64180 

3.64431 
3.64682 
3.64932 

3.65182 
3.65432 
3.65681 

7.83517 
7.84059 
7.84601 

7.85142 
7.85683 
7.86222 

7.86761 
7.87299 
7.87837 

4.90 

24.0100 

2.21359 

7.00000 

117.649 

1.69850 

3.65931 

7.88374 

4.91 
4.92 
4.93 

4.94 
4.95 
4.96 

4.97 
4.98 
4.99 

24.1081 
24.2064 
24.3049 

24.4036 
24.5025 
24.6016 

24.7009 
24.8004 
24.9001 

2.21585 
2.21811 
2.22036 

2.22261 
2.22486 
2.22711 

2.22935 
2.23159 
2.23383 

7.00714 
7.01427 
7.02140 

7.02851 
7.03562 
7.04273 

7.04982 
7.05691 
7.063<)9 

118.371 
119.095 
119.823 

120.554 
121.287 
122.024 

122.763 
123.506 
124.251 

1.69965 
1.70081 
1.70196 

1.70311 
1.70426 
1.70540 

1.706.55 
1.70769 

1.70884 

3.66179 
3.6fi428 
3.66676 

3.66924 
3.67171 
3.67418 

3.67665 
3.67911 
3.68157 

7.88909 
7.89445 
7.89979 

7.90513 
7.91046 
7.91578 

7.92110 
7.92641 
7.93171 

n] 

Powers  and  Roots 

xvii 

n 

n2 

Vn 

VlO^i 

n^ 

^ 

^100  n 

</lQn 

5.00 

25.0000 

2.23607 

7.07107 

125.000 

1.70998 

3.68403 

7.93701 

5.01 
5.02 
5.03 

5.04 
5.05 
5.06 

5.07 
5.08 
5.09 

25.1001 
25.2004 
25.3009 

25.4016 
25.5025 
25.6036 

25.7049 
25.8064 
25.9081 

2.23830 
2.24054 
2.24277 

2.24499 
2.24722 
2.24944 

2.25167 
2.?5389 
2.25610 

7.07814 
7.08520 
7.09225 

7.09930 
7.10634 
7.11337 

7.12039 
7.12741 
7.13442 

125.752 
126.506 
127.264 

128.024 

128.788 
129.554 

130.324 
131.097 
131.872 

1.71112 
1.71225 
1.71339 

1.71452 
1.71566 
1.71679 

1.71792 
1.71905 
1.72017 

3.68649 
3.68894 
3.69138 

3.69383 
3.69627 
3.69871 

3.70114 
3.70357 
3.70600 

7.94229 
7.94757 
7.95285 

7.95811 
7.96337 
7.96863 

7.97387 
7.97911 
7.98434 

5.10 

26.0100 

2.25832 

7.14143 

132.651 

1.72130 

3.70843 

7.98957 

5.11 
5.12 
5.13 

5.14 
5.15 
5.16 

6.17 
5.18 
5.19 

26.1121 
26.2144 
26.3169 

26.4196 
26.5225 
26.6256 

26.7289 
26.8324 
26.9361 

2.26053 
2.26274 
2.26495 

2.26716 
2.26936 
2.27156 

2.27376 
2.27596 
2.27816 

7.14843 
7.15542 
7.16240 

7.16938 
7.17635 
7.18331 

7.19027 
7.19722 
7.20417 

133.433 
134.218 
135.006 

135.797 
13().591 
137.388 

188.188 
138.992 
139.798 

1.72242 
1.72355 
1.72467 

1.72579 
1.72691 
1.72802 

1.72914 
1.73025 
1.73137 

3.71085 
3.71327 
3.71569 

3.71810 
3.72051 
3.72292 

3.72532 
372772 
3.73012 

7.99479 
8.00000 
8.00520 

8.01040 
8.01559 
8.02078 

8.02596 
8.03113 
8.03629 

6.20 

27.0400 

2.28035 

7.21110 

140.608 

1.73248 

3.73251 

8.04145 

5.21 
5.22 
5.23 

5.24 
5.25 
5.26 

5.27 
5.28 
5.29 

27.1441 

27.2484 
27.3529 

27.4576 
27.5625 
27.6676 

27.7729 

27.8784 
27.9841 

2.28254 
2.28473 
2.28692 

2.28910 
2.29129 
2.29347 

2.29565 
2.29783 
2.30000 

7.21803 
7.22496 
7.23187 

7.23878 
7.24569 
7.25259 

7.25948 
7.26636 
7.27324 

141.421 
142.237 
143.056 

143.878 
144.703 
145.532 

146.363 
147.198 
148.036 

1.73359 
1.73470 
1.73580 

1.73691 
1.73801 
1.73912 

1.74022 
1.74132 
1.74242 

3.73490 
3.73729 
3.73968 

3.74206 
3.74443 
3.74681 

3.74918 
3.75155 
3.75392 

8.04660 
8.05175 
8.05689 

8.06202 
8.06714 
8.07226 

8.07737 
8.08248 
8.08758 

5.30 

28.0900 

2.30217 

7.28011 

148.877 

1.74351 

3.75629 

8.ai267 

6.31 
5.32 
5.33 

6.34 
6.35 
6.36 

6.37 
6.38 
5.39 

28.1961 
28.3024 
28.4089 

28.5156 
28.6225 
28.7296 

28.8369 
28.9444 
29.0521 

2.30434 
2.30651 
2.30868 

2.31084 
2.31301 
2.31517 

2.31733 
2.31948 
2.32164 

7.28697 
7.29383 
7.30068 

7.30753 
7.31437 
7.32120 

7.32803 
7.33485 
7.34166 

149.721 
150.569 
151.419 

152.273 
153.130 
153.991 

154.854 
155.721 
156.591 

1.74461 
1.74570 
1.74680 

1.74789 

1.74898 
1.75007 

1.75116 
1.75224 
1.75333 

3.75865 
3.76101 
3.76336 

3.76571 
3.76806 
3.77041 

3.77276 
3.77509 
3.77743 

8.09776 
8.10284 
8.10791 

8.11298 
8.11804 
8.12310 

8.12814 
8.13319 
8.13822 

6.40 

29.1600 

2.32379 

7.34847 

157.464 

1.75441 

3.77976 

8.14325 

5.41 
6.42 
6.43 

5.44 
5.45 
5.46 

6.47 
6.48 
5.49 

29.2681 
29.3764 
29.4849 

29.5936 
29.7025 
29.8116 

29.9209 
30.0304 
30.1401 

2.32594 
2.32809 
2.33024 

2.33238 
2.33452 
2.33666 

2.33880 
2.34094 
2.34307 

7.35527 
7.36206 
7.36885 

7.37564 
7.38241 
7.38918 

7.39594 
7.40270 
7.40945 

158.340 
159.220 
160.103 

160.989 
161.879 
162.771 

163.667 
164.567 
165.469 

1.75549 
1.75657 
1.75766 

1.75873 
1.75981 
1.76088 

1.76196 
1.76303 
1.76410 

3.78209 
3.78442 
3.78675 

3.78907 
3.79139 
3.79371 

3.79603 
3.79834 
3.80065 

8.14828 
8.15329 
8.15831 

8.16331 
8.16831 
8.17330 

8.17829 
8.18327 
8.18824 

xviii 

Powers  and  Roots 

[H 

n 

n^ 

^/n 

VlOn 

n^ 

<^E 

</10n 

1 

^100  n 

6.50 

30.2500 

2.34521 

7.41620 

166.375 

1.76517 

3.80295 

8.19321 

5.51 
5.52 
5.53 

5.54 
5.55 
5.56 

5.57 
5.58 
5.59 

30.3601 
30.4704 
30.5809 

30.6916 
30.8025 
30.9136 

31.0249 
31.1364 
31.2481 

2.34734 
2.34947 
2.35160 

2.35372 
2.35584 
2.35797 

2.36008 
2.36220 
2.36432 

7.42294 
7.42967 
7.43640 

7.44312 
7.44983 
7.45654 

7.46324 
7.46994 
7.47663 

167.284 
168.197 
169.112 

170.031 
170.954 
171.880 

172.809 
173.741 
174.677 

1.76624 
1.76731 
1.76838 

1.76944 
1.77051 
1.77157 

1.77263 
1.77369 
1.77475 

3.80526 
3.80756 
3.80985 

3.81215 
3.81444 
3.81673 

3.81^)02 
3.82130 
3.82358 

8.19818 
8.20313 
8.20808 

8.21303 
8.21797 
8.22290 

8.22783 
8.23275 
8.23766 

5.60 

31.3600 

2.36643 

7.48331 

175.616 

1.77581 

3.82586 

8.24257 

5.61 
5.62 
5.63 

5.64 
5.65 
5.66 

5.67 
5.68 
5.69 

31.4721 
31.5844 
31.6969 

31.8096 
31.9225 
32.0356 

32.1489 
32.2624 
32.3761 

2.36854 
2.37065 
2.37276 

2.37487 
2.37697 
2.37908 

2.38118 
2.38328 
2.38537 

7.48999 
7.49()67 
7.50333 

7.50999 
7.51665 
7.52330 

7.52994 
7.53658 
7.54321 

176.558 
177.504 
178.454 

179.406 
180.362 
181.321 

182.284 
183.250 
184.220 

1.77686 
1.77792 
1.77897 

1.78003 
1.78108 
1.78213 

1.78318 
1.78422 
1.78527 

3.82814 
3.83041 
3.83268 

3.83495 
3.83722 
3.83948 

3.84174 
3.843^)9 
3.84625 

8.24747 
8.25237 
8.25726 

8.26215 
8.26703 
8.27190 

8.27677 
8.28164 
8.28649 

5.70 

32.4900 

2.38747 

7.54983 

185.193 

1.78632 

3.84850 

8.29134 

5.71 
5.72 
5.73 

5.74 
5.75 
5.76 

5.77 

5.78 
5.79 

32.6041 
32.7184 
32.8329 

32.9476 
33.0625 
33.1776 

33.2929 
33.4084 
33.5241 

2.38956 
2.39165 
2.39374 

2.39583 
2.39792 
2.40000 

2.40208 
2.40416 
2.40624 

7.55645 
7.56307 
7.56968 

7.57628 
7.58288 
7.58947 

7.59605 
7.(50263 
7.(50<>20 

186.1(59 
187.149 
188.133 

189.119 
irK).109 
191.103 

192.100 
193.101 
194.105 

1.78736 
1.78840 
1.78944 

1.79048 
1.79152 
1.79256 

1.79360 
1.79463 
1.79567 

3.85075 
3.85300 
3.85524 

3.85748 
3.85972 
3.86196 

3.86419 
3.86642 
3.868(55 

8.2<)619 
8.30103 
8.30587 

8.31069 
8.31552 
8.32034 

8.32515 
8.32995 
8.33476 

5.80 

33.6400 

2.40832 

7.61577 

195.112 

1.7i¥570 

3.87088 

8.33955 

5.81 
5.82 
5.83 

5.84 
5.85 
5.86 

5.87 
5.88 
5.89 

33.7561 
33.8724 
33.9889 

34.1056 
34.2225 
34.3396 

34.4569 
34.5744 
34.6921 

2.41039 
2.41247 
2.41454 

2.41661 
2.41868 
2.42074 

2.42281 
2.42487 
2.42693 

7.62234 
7.62889 
7.63544 

7.64199 
7.64853 
7.65506 

7.66159 
7.66812 
7.67463 

196.123 
197.137 
198.155 

199.177 
200.202 
201.230 

202.262 
203.297 
204.3.36 

1.79773 
1.79876 
1.79979 

1.80082 
1.80185 
1.80288 

1.80390 
1.80492 
1.80595 

3.87310 
3.87532 
3.87754 

3.87975 
3.88197 
3.88418 

3.88639 
3.88859 
3.89080 

8.34434 
8.34913 
8.35390 

8.35868 
8.36345 
8.36821 

8.37297 
8.37772 
8.38247 

5.90 

34.8100 

2.42899 

7.(>8115 

205.379 

1.80697 

3.89;X)0 

8.38721 

5.91 
5.92 
5.93 

5.94 
5.95 
5.96 

5.97 
5.98 
5.99 

34.9281 
35.0464 
35.1649 

35.283fi 
35.4025 
35.5216 

a5.6409 
.35.7604 
35.8801 

2.43105 
2.43311 
2.43516 

2.43721 
2.43926 
2.44131 

2.44336 
2.44540 
2.44745 

7.68765 
7.69415 
7.70065 

7.70714 
7.71362 
7.72010 

7.72658 
7.7a305 
7.73951 

206.425 
207.475 
208.528 

209.585 
210.(545 
211.709 

212.776 
213.847 
214.V)22 

1.80799 
1.80i)01 
1.81003 

1.81104 
1.81206 
1.81307 

1.81409 
1.81510 
1.81611 

3.89519 
3.89739 
3.89958 

3.90177 
3.90396 
3.90615 

3.90833 
3.91051 
3.912(59 

8.39194 
8.39667 
8.40140 

8.40612 
8.41083 
8.41554 

8.42025 
8.42494 
8.42964 

11] 

Powers  and  Roots 

xix 

n 

n^ 

V^ 

V10»* 

n^ 

</^ 

^10  n 

i/lOOn 

6.00 

36.0000 

2.44949 

7.74597 

216.000 

1.81712 

3.91487 

8.43433 

6.01 
6.02 
6.03 

6.04 
6.05 
6.06 

6.07 
6.08 
6.09 

36.1201 
36.2404 
36.3609 

36.4816 
36.6025 
36.7236 

36.8449 
36.9664 
37.0881 

2.45153 
2.45357 
2.45561 

2.45764 
2.45967 
2.46171 

2.46374 
2.40577 
2.46779 

7.75242 

7.75887 
7.76531 

7.77174 

7.77817 
7.78460 

7.79102 
7.79744 

7.80385 

217.082 
218.167 
219.256 

220.349 
221.445 
222.545 

223.649 
224.756 

225.867 

1.81813 
1.81914 
1.82014 

1.82115 
1.82215 
1.82316 

1.82416 
1.82516 
1.82616 

3.91704 
3.91921 
3.92138 

3.92355 
3.92571 
3.92787 

3.93003 
3.93219 
3.934.34 

8.43901 
8.44369 
8.44836 

8.45303 
8.45769 
8.46235 

8.46700 
8.47165 
8.47629 

6.10 

37.2100 

2.46982 

7.81025 

226.981 

1.82716 

3.93650 

8.48093 

6.11 
6.12 
6.13 

6.14 
6.15 
6.16 

6.17 
6.18 
6.19 

37.3321 
37.4544 
37.5769 

37.6996 
37.8225 
37.9456 

38.0689 
38.1924 
38.3161 

2.47184 
2.47386 
2.47588 

2.47790 
2.47992 
2.48193 

2.48395 
2.48596 
2.48797 

7.81665 
7.82304 
7.82943 

7.83582 
7.84219 

7.84857 

7.85493 
7.86130 
7.86766 

228.099 
229.221 
230.346 

231.476 
232.608 
233.745 

234.885 
230.029 
237.177 

1.82816 
1.82915 
1.83015 

1.83115 
1.83214 
1.83313 

1.83412 
1.83511 
1.83610 

3.93865 
3.94079 
3.94294 

3.91508 
3.94722 
3.94936 

3.95150 
3.95363 
3.95576 

8.48556 
8.49018 
8.49481 

8.49942 
8.50403 
8.50864 

8.51324 
8.51784 
8.52243 

6.20 

38.4400 

2.48998 

7.87401 

238.328 

1.83709 

3.95789 

8.52702 

6.21 
6.22 
6.23 

6.24 
6.25 
6.26 

6.27 
6.28 
6.29 

38.5641 
38.6884 
38.8129 

38.9376 
39.0625 
39.1876 

39.3129 
39.4384 
39.5(i41 

2.49199 
2.49399 
2.49600 

2.49800 
2.50000 
2.50200 

2.50400 
2.50599 
2.50799 

7.88036 
7.88670 
7.89303 

7.89937 
7.90569 
7.91202 

7.91833 
7.92465 
7.93095 

239.483 
240.642 
241.804 

242.971 
244.141 
245.314 

246.492 

247.673 
248.858 

1.83808 
1.83<K)6 
1.84005 

1.84103 
1.84202 
1.&4300 

1.84398 
1.844()6 
1.84594 

3.96002 
3.90214 
3.96427 

3.966.38 
3.96850 
3.97062 

3.97273 

3.97484 
3.97695 

8.53160 
8.53618 
8.54075 

8.54532 
8.54988 
8.55444 

8.55899 
8.56a54 
8.56808 

6.30 

39.6900 

2.50<)98 

7.9.3725 

250.047 

1.84691 

3.97906 

8.57262 

6.31 
6.32 
6.33 

6.34 
6.35 
6.36 

6.37 
6.38 
6.39 

39.8161 
39.9424 
40.0689 

40.1956 
40.3225 
40.4496 

40.5769 
40.7044 
40.8321 

2.51197 
2.51396 
2.51595 

2.51794 
2.51992 
2.52190 

2.52389 
2.52587 
2.52784 

7.94^355 
7.94984 
7.95613 

7.96241 
7.%869 
7.97496 

7.98123 

7.98749 
7.99375 

251.240 
252.4,% 
253.636 

254.840 
256.048 
257.259 

258.475 
259.694 
260.917 

1.84789 
1.84887 
1.84984 

1.85082 
1.85179 
1.85276 

1.85373 
1.85470 
1.8.5567 

3.98116 
3.98326 
3.98536 

3.98746 
3.98956 
3.99165 

3.99374 
3.99583 
3.99792 

8.57715 
8.58168 
8.58620 

8.59072 
8.59524 
8.59975 

8.60425 
8.60875 
8.61.325 

6.40 

40.9600 

2.52982 

8.00000 

262.144 

1.85664 

4.00000 

8.61774 

6.41 
6.42 
6.43 

6.44 
6.45 
6.46 

6.47 
6.48 
6.49 

41.0881 
41.2164 
41.3449 

41.4736 
41.6025 
41.7316 

41.8609 
41.9904 
42.1201 

2.53180 
2.53,377 
2.53574 

2.53772 
2.53969 
2.54165 

2.54362 
2.54.558 
2.54755 

8.00625 
8.01249 
8.01873 

8.02496 
8.03119 
8.03741 

8.04363 
8.04984 
8.0.5605 

263.375 
264.609 
265.848 

267.090 
268.336 
269.586 

270.840 
272.098 
273.359 

1.8.5760 
1.85857 
1.85953 

1.86050 
1.86146 
1.86242 

1.86338 
1.864.34 
1.865;^ 

4.00208 
4.()041(> 
4.00624 

4.00832 
4.01039 
4.01246 

4.01453 
4.01660 
4.01866 

8.62222 
8.62671 
8.63118 

8.63566 
8.64012 
8.64459 

8.64904 
8.65350 
8.65795 

XX 


Powers  and  Roots 


[n 


n 

n^ 

Vn 

VIO^ 

n3 

^n 

i/lOn 

\ 

</100n 

6.50 

42.2.500 

2.54951 

8.06226 

274.625 

1.86626 

4.02073 

8.66239 

6.51 
6.52 
6.53 

6.54 
6.55 
6.56 

6.57 
6.58 
6.59 

42..3801 
42.5104 
42.6409 

42.7716 
42.t)025 
43.0336 

43.1649 
43.2964 
43.4281 

2.55147 
2.55343 
2.55539 

2.55734 
2.55930 
2.56125 

2.56320 
2.56515 
2.56710 

8.06846 
8.07465 
8.08084 

8.08703 
8.09321 
8.09938 

8.10555 
8.11172 
8.11788 

275.894 
277.168 
278.445 

279.726 
281.011 
282.300 

283.593 

284.890 
286.191 

1.86721 
1.86817 
1.86912 

1.87008 
1.87103 
1.87198 

1.87293 

1.87388 
1.87483 

4.02279 
4.02485 
4.02690 

4.02896 
4.03101 
4.03306 

4.03511 
4.03715 
4.03920 

8.66683 
8.67127 
8.67570 

8.68012 
8.68455 
8.68896 

8.69338 
8.69778 
8.70219 

6.60 

43.5600 

2.56905 

8.12404 

287.496 

1.87578 

4.04124 

8.70659 

6.61 
6.62 
6.63 

6.64 
6.65 
6.66 

6.67 
6.68 
6.69 

43.6921 
43.8244 
43.9569 

44.0896 
44.2225 
44.3556 

44.4889 
44.6224 
44.7561 

2.57099 
2.57294 

2.57488 

2.57682 
2.57876 
2.58070 

2.58263 
2.58457 
2.58650 

8.13019 
8.13634 
8.14248 

8.14862 
8.15475 
8.16088 

8.16701 
8.17313 
8.17924 

288.805 
290.118 
291.434 

292.755 
294.080 
295.408 

296.741 
298.078 
299.418 

1.87672 
1.87767 
1.87862 

1.87956 
1.88050 
1.88144 

1.88239 
1.88333 

1.88427 

4.04328 
4.04532 
4.04735 

4.04939 
4.05142 
4.05345 

4.05548 
4.05750 
4.05953 

8.71098 
8.71537 
8.71976 

8.72414 
8.72852 
8.73289 

8.73726 
8.74162 
8.74598 

6.70 

44.8900 

2.58844 

8.18535 

300.763 

1.88520 

4.06155 

8.75034 

6.71 
6.72 
6.73 

6.74 
6.75 
6.76 

6.77 
6.78 
6.79 

45.0241 
45.1584 
45.2929 

45.4276 
45.5625 
45.6976 

45.8329 
45.9684 
46.1041 

2.59037 
2.59230 
2.59422 

2.59615 
2.59808 
2.60000 

2.60192 
2.60384 
2.60576 

8.19146 
8.19756 
8.20366 

8.20975 
8.21584 
8.22192 

8.22800 
8.23408 
8.24015 

302.112 
303.464 
304.821 

306.182 
307.547 
308.916 

310.289 
311.666 
313.047 

1.88614 
1.88708 
1.88801 

1.88895 
1.88988 
1.89081 

1.89175 
1.89268 
1.89361 

4.06357 
4.06559 
4.06760 

4.06961 
4.07163 
4.07364 

4.07564 
4.07765 
4.07965 

8.75469 
8.75904 
8.76338 

8.76772 
8.77205 
8.77638 

8.78071 
8.78503 
8.78935 

6.80 

46.2400 

2.60768 

8.24621 

314.432 

1.89454 

4.08166 

8.79366 

6.81 
6.82 
6.83 

6.84 
6.85 
6.86 

6.87 
6.88 
6.89 

46.3761 
46.5124 
46.6489 

46.7856 
46.9225 
47.0596 

47.1969 
47.3344 

47.4721 

2.60960 
2.61151 
2.61343 

2.61534 
2.61725 
2.61916 

2.62107 

2.62298 
2.62488 

8.25227 
8.25833 
8.26438 

8.27043 
8.27647 
8.28251 

8.28855 
8.29458 
8.30060 

315.821 
317.215 
318.612 

320.014 
321.419 
322.829 

324.243 
325.661 
327.083 

1.89546 
1.89639 
1.89732 

1.89824 
1.89917 
1.90009 

1.90102 
1.90194 
1.90286 

4.08365 
4.08565 
4.08765 

4.08964 
4.09163 
4.09362 

4.09561 
4.09760 
4.09958 

8.79797 
8.80227 
8.80657 

8.81087 
8.81516 
8.81945 

8.82373 
8.82801 
8.83228 

6.90 

47.6100 

2.62679 

8.30662 

328.509 

1.90378 

4.10157 

8.83()56 

6.91 
6.92 
6.93 

6.94 
6.95 
6.96 

6.97 
6.98 
6.99 

47.7481 
47.8864 
48.0249 

48.1636 
48..3025 
48.4416 

48.5809 
48.7204 
48.8601 

2.62869 
2.63059 
2.63249 

2.63439 
2.63629 
2.63818 

2.64008 
2.64197 
2.64386 

8.31264 
8.31865 
8.32466 

8.33067 
8.33667 
8.34266 

8.34865 
8.35464 
8.36062 

329.939 
331.374 
332.813 

3.34.255 
335.702 
337.154 

338.609 
340.068 
341.532 

1.90470 
1.90562 
1.90653 

1.90745 
1.90837 
1.90928 

1.91019 
1.91111 
1.91202 

4.10355 
4.10552 
4.10750 

4.10948 
4.11145 
4.11342 

4.11539 
4.11736 
4.11932 

8.84082 
8.84509 
8.84934 

8.85360 
8.85785 
8.86210 

8.86634 
8.87058 
8.87481 

H] 

Powers  and  Roots 

xxi 

n 

n^ 

y/n 

y/lOn 

n^ 

^ 

</lOn 

^100  w 

7.00 

49.0000 

2.64575 

8.36660 

343.000 

1.91293 

4.12129 

8.87904 

7.01 
7.02 
7.03 

7.04 
7.05 
7.06 

7.07 
7.08 
7.09 

49.1401 
49.2804 
49.4209 

49.5616 
49.7025 
49.8436 

49.9849 
50.1264 
50.2681 

2.64764 
2.64953 
2.65141 

2.65330 
2.65518 
2.65707 

2.65895 
2.66083 
2.66271 

8.37257 
8.37854 
8.38451 

8.39047 
8.39643 
8.40238 

8.40833 
8.41427 
8.42021 

:344.472 
345.948 
347.429 

348.914 
350.403 
351.896 

353.393 
354.895 
3.'6.401 

1.91384 
1.91475 
1.91566 

1.91657 
1.91747 
1.91838 

1.91929 

.1.92019 

1.92109 

4.12325 
4.12521 
4.12716 

4.12912 
4.13107 
4.13303 

4.13498 
4.13693 
4.13887 

8.88327 
8.88749 
8.89171 

8.89592 
8.90013 
8.90434 

8.90854 
8.91274 
8.91693 

7.10 

7.11 
7.12 
7.13 

7.14 
7.15 
7.16 

7.17 
7.18 
7.19 

50.4100 

2.66458 

8.42615 

357.911 

1.92200 

4.14082 

8.92112 

50.5521 
50.6944 
50.8369 

50.9796 
51.1225 
51.2656 

51.4089 
51.5524 
51.6961 

2.66(546 
2.6(5833 
2.67021 

2.67208 
2.67395 
2.67582 

2.67769 
2.67955 
2.68142 

8.43208 
8.43801 
8.44393 

8.44985 
8.45577 
8.46168 

8.46759 
8.47349 
8.47939 

359.425 
360.944 
362.467 

363.994 
365.526 
367.062 

368.602 
370.146 
371.695 

1.92290 
1.92380 
1.92470 

1.92560 
1.92650 
1.92740 

1.92829 
1.92919 
1.93008 

4.14276 
4.14470 
4.14664 

4.14858 
4.15052 
4.15245 

4.15438 
4.ir)631 
4.15824 

8.92531 
8.92949 
8.93367 

8.93784 
8.94201 
8.94618 

8.95034 
8.95450 
8.95866 

7.20 

51.8400 

2.68328 

8.48528 

373.248 

1.93098 

4.16017 

8.96281 

7.21 
7.22 
7.23 

7.24 
7.25 
7.26 

7.27 
7.28 
7.29 

51.9841 
52.1284 
52.2729 

52.4176 
52.5625 
52.7076 

52.8529 
52.9984 
53.1441 

2.68514 
2.68701 
2.68887 

2.69072 
2.69258 
2.69444 

2.69629 
2.69815 
2.70000 

8.49117 
8.49706 
8.50294 

8.50882 
8.51469 
8.52056 

8.52643 
8.53229 
8.53815 

374.805 
376.367 
377.933 

379.503 
381.078 
382.657 

384.241 

385.828 
387.420 

1.93187 
1.93277 
1.93366 

1.93455 
1.93544 
1.93633 

1.93722 
1.93810 
1.93899 

4.1(5209 
4.16402 
4.16594 

4.16786 
4.1(5978 
4.17169 

4.17361 
4.17552 
4.17743 

8.96696 
8.97110 
8.97524 

8.97938 
8.98351 
8.98764 

8.99176 
8.99588 
9.00000 

7.30 

53.2900 

2.70185 

8.54400 

389.017 

1.93988 

4.17934 

9.00411 

7.31 
7.32 
7.33 

7.34 
7.35 
7.36 

7.37 
7.38 
7.39 

53.4361 
53.5824 
53.7289 

53.8756 
54.0225 
54.1696 

54.3169 
54.4644 
54.6121 

2.70370 
2.70555 
2.70740 

2.70924 
2.71109 
2.71293 

2.71477 
2.71662 

2.71846 

8.54985 
8.55570 
8.56154 

8.56738 
8.57321 
8.57904 

8.58487 
8.59069 
8.59651 

390.618 
392.223 
393.833 

395.447 
397.065 
398.688 

400.316 
401.947 
403.583 

1.94076 
1.94165 
1.94253 

1.94341 
1.94430 
1.94518 

1.94606 
1.94(594 
1.94782 

4.18125 
4.18315 
4.18506 

4.18696 
4.18886 
4.19076 

4.19266 
4.19455 
4.19644 

9.00822 
9.01233 
9.01643 

9.02053 
9.02462 
9.02871 

9.03280 
9.03689 
9.04097 

7.40 

54.7600 

2.72029 

8.602,33 

405.224 

1.94870 

4.19834 

9.04504 

7.41 
7.42 
7.43 

7.44 
7.45 
7.46 

7.47 
7.48 
7.49 

54.9081 
55.0564 
55.2049 

55.3536 
55.5025 
55.6516 

55.8009 
55.9504 
56.1001 

2.72213 

2.72.397 
2.72580 

2.72764 
2.72947 
2.73130 

2.73313 
2.73496 
2.73679 

8.60814 
8.61394 
8.61974 

8.62554 
8.631  :h 
8.63713 

8.64292 
8.64870 
8.65448 

406.869 
408.518 
410.172 

411.831 
413.494 
415.161 

416.833 
418.509 
420.190 

1.94957 
1.95045 
1.95132 

1.95220 
1.95307 
1.95.395 

1.95482 
1.955(59 
1.95656 

4.20023 
4.20212 
4.20400 

4.20589 
4.20777 
4.20965 

4.21153 
4.21341 
4.21529 

9.04911 
9.05318 
9.05725 

9.06131 
9.06537 
9.06942 

9.07347 
9.07752 
9.08156 

xxii 

Powers  and  Roots 

Cn 

n 

n^ 

Vn 

VlOw 

n^ 

</n 

^10  n 

</100n 

7.50 

56.2500 

2.73861 

8.66025 

421.875 

1.95743 

4.21716 

9.08560 

7.51 
7.52 
7.53 

7.54 
7.55 
7.56 

7.57 
7.58 
7.59 

56.4001 
56.5504 
56.7009 

56.8516 
57.0025 
57.1536 

57.3049 
57.4564 
57.6081 

2.74044 
2.74226 

2.74408 

2.74591 
2.74773 
2.74955 

2.75136 
2.75318 
2.75500 

8.()6(303 
8.()7179 
8.67756 

8.68332 
8.68907 
8.69483 

8.70057 
8.70632 
8.71206 

423.565 
425.259 
426.958 

428.661 
430.369 
432.081 

433.798 
435.520 
437.245 

1.95830 
1.95917 
1.96004 

1.96091 
1.96177 
1.96264 

1.96350 
1.9(5437 
l.%523 

4.21904 
4  22091 
4.22278 

4.22465 
4.22651 
4.22838 

4.23024 
4.23210 
4.23396 

9.08%4 
9.09367 
9.09770 

9.10173 
9.10575 
9.10977 

9.11378 
9.11779 
9.12180 

7.60 

57.7600 

2.75681 

8.71780 

438.976 

1.{K5610 

4.23582 

9.12581 

7.61 
7.62 
7.63 

7.64 
7.65 
7.66 

7.67 
7.68 
7.69 

57.9121 
58.0644 
58.2169 

58.3696 
58.5225 
58.6756 

58.8289 
58.9824 
59.1361 

59.2900 

2.75862 
2.76043 
2.76225 

2.76405 
2.76586 
2.76767 

2.76948 
2.77128 
2.77308 

8.72353 
8.72926 
8.73499 

8.74071 
8.74643 
8.75214 

8.75785 
8.76356 
8.7692G 

440.711 
442.451 
444.195 

445.944 
447.697 
449.455 

451.218 
452.985 
454.757 

1.9(5696 
1.96782 
1.96868 

1.96954 
1.97040 
1.97126 

1.97211 
1.97297 
1.97383 

4.23768 
4.23954 
4.24139 

4.24324 
4.24509 
4.24694 

4.24879 
4.25063 
4.25248 

9.12981 
9.13380 
9.13780 

9.14179 
9.14577 
9.14976 

9.15,374 
9.15771 

9.16169 

7.70 

2.77489 

8.77496 

456.533 

1.97468 

4.25432 

9.16566 

7.71 
7.72 
7.73 

7.74 
7.75 
7.76 

7.77 
7.78 
7.79 

59.4441 
59.5984 
59.7529 

59.9076 
60.0625 
60.2176 

60.3729 
60.5284 
60.6841 

2.77069 
2.77849 
2.78029 

2.78209 
2.78388 
2.78568 

2.78747 
2.78927 
2.79106 

8.78066 
8.78635 
8.79204 

8.79773 
8.80341 
8.80909 

8.81476 
8.82043 
8.82610 

458.314 
460.100 
461.890 

463.685 
465.484 
467.289 

469.097 
470.911 

472.729 

1.97554 
1.97639 
1.97724 

1.97809 
1.97895 
1.97980 

1.980G5 
1.98150 
1.98234 

4.25616 
4.25800 
4.25984 

4.26167 
4.26351 
4.26534 

4.26717 
4.26900 

4.27083 

9.16962 
9.17359 
9.17754 

9.18150 
9.18545 
9.18940 

9.19336 
9.19729 
9.20123 

7.80 

60.8400 

2.79285 

8.83176 

474.552 

1.98319 

4.27266 

9.20516 

7.81 
7.82 
7.83 

7.84 
7.85 
7.86 

7.87 
7.88 
7.89 

60.9961 
61.1524 
61.3089 

61.4656 
61.6225 
61.7796 

61.93(39 
62.0944 
62.2521 

2.79464 
2.79643 
2.79821 

2.80000 
2.80179 
2.80357 

2.80535 
2.80713 
2.80891 

8.83742 
8.84308 
8.84873 

8.85438 
8.86002 
8.86566 

8.87130 
8.876M 
8.88257 

476.380 
478.212 
480.049 

481.890 
483.737 
485.588 

487.443 
489.304 
491.1(3i) 

1.98404 
1.98489 
1.98573 

1.98658 
1.98742 
1.98826 

1.98911 
1.98995 
1.9<)079 

4.27448 
4.27631 
4.27813 

4.27995 
4.28177 
4.28359 

4.28540 
4.28722 
4.28903 

9.20910 
9.21302 
9.21695 

9.22087 
9.22479 

9.22871 

9.23262 
9.23653 
9.24043 

7  90 

62.4100 

2.810()9 

8.88819 

493.039 

1.<M)H)3 

4.29084 

9.24434 

7.91 
7.92 
7.93 

7.94 
7.95 
7.96 

7.97 
7.98 
7.99 

62.5681 
62.7264 
62.8849 

63.0436 
63.2025 
63.3616 

63.5209 
63.6804 
63.8401 

2.81247 
2.81425 
2.81603 

2.81780 
2.81957 
2.82135 

2.82312 
2.82489 
2.826(56 

8.89382 
8.895)44 
8.90505 

8.910(37 
8.91628 
8.92188 

8.92749 
8.93308 
8.93868 

494.914 
496.793 
498.677 

500.566 
502.4(50 
504.358 

506.262 
508.170 
510.082 

1.99247 
1.99331 
1.99415 

1.994<)9 
1.99582 
1.99066 

1.99750 
1.99833 
1.99917 

4.29265 
4.2^>446 
4.29627 

4.29807 

4.2<^)987 
4.30168 

4.30348 
4.30528 
4.;W07 

9.24823 
9.25213 
9.25602 

9.25991 
9.26380 
9.26768 

9.27156 
9.27544 
9.27931 

n] 


Powers  and  Boots 


XXlll 


n 

W,2 

Vn 

VIO^ 

n^ 

^ 

^10^ 

</100n 

8.00 

64.0000 

2.82843 

8.94427 

512.000 

2.00000 

4.30887 

9.28318 

8.01 
8.02 
8.03 

64.1(501 
64.3204 
64.4809 

2.83019 
2.83196 
2.83373 

8.94986 
8.95545 
8.96103 

513.922 
515.850 
517.782 

2.00083 
2.00167 
2.00250 

4.31066 
4.31246 
4.31425 

9.28704 
9.29091 
9.29477 

8.04 
8.05 
8.06 

64.6416 
64.8025 
64.9636 

2.83549 
2.83725 
2.83901 

8.96660 
8.97218 
8.97775 

519.718 
521.660 
523.607 

2.00333 
2.00416 
2.00499 

4.31604 
4.31783 
4.31961 

9.29862 
9.30248 
9.30633 

8.07 
8.08 
8.09 

65.1249 
65.2864 
65.4481 

2.84077 
2.84253 
2.84429 

8.98332 
8.98888 
8.99444 

525.558 
527.514 
529.475 

2.00582 
-2.00664 
2.00747 

4.32140 
4.32318 
4.32497 

9.31018 
9.31402 
9.31786 

8.10 

65.6100 

2.84605 

9.00000 

531.441 

2.00830 

4.32675 

9.32170 

8.11 
8.12 
8.13 

65.7721 
65.9344 
66.0969 

2.84781 
2.84956 
2.85132 

9.00555 
9.01110 
9.01665 

533.412 
535.387 
537.368 

2.00912 
2.00995 
2.01078 

4.32853 
4.33031 
4.33208 

9.32553 
9.32936 
9.33319 

8.14 
8.15 
8.16 

66.2596 
66.4225 
66.5856 

2.85307 
2.85482 
2.85657 

9.02219 
9.02774 
9.03327 

539.353 
541.343 
543.338 

2.01160 
2.01242 
2.01325 

4.33386 
4.33563 
4.33741 

9.33702 
9.34084 
9.34466 

8.17 
8.18 
8.19 

66.7489 
66.9124 
67.0761 

2.85832 
2.86007 
2.86182 

9.03881 
9.04434 
9.04986 

545.3.39 
547.343 
549.353 

2.01407 
2.01489 
2.01571 

4.33918 
4.34095 
4.34271 

9.34847 
9.35229 
9.35610 

8.20 

67.2400 

2.86356 

9.05539 

551.368 

2.01653 

4.34448 

9.35990 

8.21 
8.22 
8.23 

67.4041 
67.5684 
67.7329 

2.86531 
2.86705 
2.86880 

9.06091 
9.06642 
9.07193 

553.388 
555.412 
557.442 

2.01735 
2.01817 
2.01899 

4.34625 
4.34801 
4.34977 

9.36370 
9.36751 
9.37130 

8.24 
8.25 
8.26 

67.8976 
68.0625 
68.2276 

2.87054 
2.87228 
2.87402 

9.07744 
9.08295 
9.08845 

559.476 
561.516 
563.560 

2.01980 
2.020(>2 
2.02144 

4.35153 
4.35329 
4.35505 

9.37510 
9.37889 
9.38268 

8.27 
8.28 
8.29 

68.3929 
68.5584 
68.7241 

2.87576 
2.87750 
2.87924 

9.09395 
9.09945 
9.10494 

565.609 
567. (J64 
569.723 

2.02225 
2.02307 
2.02388 

4.35681 
4.35856 
4.36032 

9.38646 
9.39024 
9.39402 

8.30 

68.8900 

2.88097 

9.11043 

571.787 

2.02469 

4.36207 

9.39780 

8.31 
8.32 
8.33 

69.0561 
69.2224 
69.3889 

2.88271 
2.88144 
2.88617 

9.11592 
9.12140 
9.12688 

573.856 
575.930 
578.010 

2.02551 
2.02632 
2.02713 

4.36382 
4.36557 
4.36732 

9.40157 
9.40534 
9.40911 

8.34 
8.35 
8.36 

69.5556 
69.7225 
69.8896 

2.88791 
2.88964 
2.89137 

9.13236 
9.13783 
9.14330 

580.094 
582.183 
584.277 

2.02794 

2.02875 
2.02956 

4.36907 
4.37081 
4.37256 

9.41287 
9.41663 
9.42039 

8.37 
8.38 
8.39 

70.0569 
70.2244 
70.3921 

2.89310 

2.89482 
2.89655 

9.14877 
9.15423 
9.15969 

586.376 
588.480 
590.590 

2.03037 
2.03118 
2.03199 

4.37430 
4.37604 
4.37778 

9.42414 
9.42789 
9.43164 

8.40 

70.5600 

2.89828 

9.16515 

592.704 

2.03279 

4.37952 

9.43539 

8.41 
8.42 
8.43 

70.7281 
70.8964 
71.0649 

2.90000 
2.90172 
2.90345 

9.17061 
9.17606 
9.18150 

591.823 
59(3.948 
599.077 

2.03360 
2.03440 
2.03521 

4.38126 
4.38299 
4.38473 

9.43913 
9.44287 
9.44661 

8.44 
8.45 
8.46 

71.2336 
71.4025 
71. .5716 

2.90517 
2.90689 
2.90861 

9.18695 
9.19239 
9.19783 

601.212 
603.351 
605.496 

2.03601 
2.03682 
2.03762 

4.-38646 
4.38819 
4.38992 

9.45034 
9.45407 
9.45780 

8.47 
8.48 
8.49 

71.7409 
71.9104 
72.0801 

2.91033 
2.91204 
2.91376 

9.20326 
9.20869 
9.21412 

607.645 
609.800 
611.960 

2.03842 
2.03923 
2.04003 

4.39165 
4.39338 
4.39510 

9.46152 
9.46525 

9.46897 

xxiv 

Powers  and  Roots 

[11 

n 

W2 

y/n 

VlOw 

n3 

</n 

^10  n 

S/100»i 

8.50 

8.51 
8.52 
8.53 

8.54 
8.55 
8.56 

8.57 
8.58 
8.59 

72.2500 

2.91548 

9.21954 

614.125 

2.04083 

4.39683 

9.47268 

72.4201 
72.5904 
72.7609 

72.9316 
73.1025 
73.2736 

73.4449 
73.6164 
73.7881 

2.91719 
2.91890 
2.92062 

2.92233 
2.92404 
2.92575 

2.9274(3 
2.92916 
2.93087 

9.22497 
9.23038 
9.23580 

9.24121 
9.246(>2 
9.25203 

9.25743 
9.26283 
9.26823 

616.295 
618.470 
620.650 

622.836 
625.026 
627.222 

629.423 
631.629 
633.840 

2.04163 
2.04243 
2.04323 

2.04402 
2.04482 
2.04562 

2.04641 
2.04721 
2.04801 

4.39855 
4.40028 
4.40200 

4.40372 
4.40543 
4.40715 

4.40887 
4.41058 
4.41229 

9.47640 
9.48011 
9.48381 

9.48752 
9.49122 
9.49492 

9.49861 
9.50231 
9.50600 

8.60 

73.9600 

2.93258 

9.27362 

636.05(5 

2.04880 

4.41400 

9..50t)69 

8.61 
8.62 
8.63 

8.64 
8.65 
8.66 

8.67 
8.68 
8.69 

74.1321 
74.3044 
74.4769 

74.6496 
74.8225 
74.9956 

75.1689 
75.3424 
75.5161 

2.93428 
2.93598 
2.93709 

2.93939 
2.94109 
2.94279 

2.94449 
2.94618 
2.94788 

9.27901 
9.28440 
9.28978 

9.29516 
9.30054 
9.30591 

9.31128 
9.31665 
9.32202 

638.277 
640.504 
642.736 

644.973 
647.215 
649.462 

651.714 
653.972 
656.235 

2.04959 
2.0r.039 
2.05118 

2.05197 
2.05276 
2.05355 

2.05434 
2.05513 
2.05592 

4.41571 
4.41742 
4.41913 

4.42084 
4.42254 
4.42425 

4.42595 
4.42765 
4.42935 

9.51337 
9.51705 
9.52073 

9.52441 

9.52808 
9.53175 

9.53542 
9.53908 
9.54274 

8.70 

75.6900 

2.f)4958 

9.32738 

658.503 

2.05671 

4.43105 

9.54(540 

8.71 
8.72 
8.73 

8.74 
8.75 
8.76 

8.77 
8.78 
8.79 

75.8641 
76.0384 
76.2129 

76.3876 
76.5625 
76.7376 

76.9129 

77.0884 
77.2641 

2.95127 
2.95296 
2.95466 

2.95635 
2.95804 
2.95973 

2.96142 
2.96311 
2.96479 

9.33274 
9.33809 
9.34345 

9.34880 
9.35414 
9.35949 

9.36483 
9.37017 
9.37550 

660.776 
663.055 
665.339 

667.628 
669.922 
672.221 

674.526 
(576.836 
679.151 

2.05750 
2.05828 
2.05907 

2.05986 
2.06064 
2.06143 

2.06221 
2.06299 
2.06378 

4.43274 
4.43444 
4.43613 

4.43783 
4.43952 
4.44121 

4.44290 
4.44459 
4.44627 

9.55006 
9.55371 
9.55736 

9.56101 
9.56466 
9.56830 

9.57194 
9.57557 
9.57921 

8.80 

77.4400 

2.96648 

9.38083 

681.472 

2.06456 

4.44796 

9.58284 

8.81 
8.82 
8.83 

8.84 
8.85 
8.86 

8.87 
8.88 
8.89 

77.6161 
77.7924 
77.9689 

78.1456 
78.3225 
78.4996 

78.6769 
78.8544 
79.0321 

2.96816 
2.96985 
2.97153 

2.97321 
2.97489 
2.97658 

2.97825 
2.97993 
2.98161 

9.38616 
9.39149 
9.39681 

9.40213 
9.40744 
9.41276 

9.41807 
9.42338 
9.42868 

683.798 
686.129 
688.465 

690.807 
693.154 
695.506 

697.864 
700.227 
702.595 

2.06534 
2.06(512 
2.06690 

2.06768 
2.06846 
2.06924 

2.07002 
2.07080 
2.071.57 

4.44964 
4.45133 
4.45301 

4.45469 
4.45(537 
4.45805 

4.45972 
4.46140 
4.46307 

9.58(547 
9.59009 
9.59372 

9.59734 
9.60095 
9.60457 

9.60818 
9.61179 
9.61540 

8.90 

79.2100 

2.98329 

9.43398 

704.969 

2.072:55 

4.46475 

9.61900 

8.91 
8.92 
8.93 

8.94 
8.95 
8.96 

8.97 
8.98 
8.99 

79.3881 
79.5664 
79.7449 

79.92.36 
80.1025 
80.2816 

80.4609 
80.6404 
80.8201 

2.98496 
2.98664 
2.98831 

2.98998 
2.99166 
2.99333 

2.99500 
2.99(566 
2.998.33 

9.43928 
9.44458 
9.44987 

9.45516 
9.4(K)44 
9.46573 

9.47101 
9.47629 
9.48156 

707.348 
709.732 
712.122 

714.517 
716.917 
719.323 

721.734 
724.151 
726.573 

2.07313 
2.073f)0 
2.07468 

2.07545 
2.07622 
2.07700 

2.07777 
2.07854 
2.07931 

4.46642 
4.46809 
4.46976 

4.47142 
4.47309 
4.47476 

4.47642 
4.47808 
4.47974 

9.62260 
9.(52620 
9.62980 

9.63339 
9.(5;?698 
9.64057 

9.64415 
9.64774 
9.65132 

11] 

Powers  and  Roots 

XXV 

n 

n^ 

v^ 

VlOri 

n^ 

^ 

^10  n 

^100^ 

9.00 

81.0000 

3.00000 

9.48683 

729.000 

2.08008 

4.48140 

9.65489 

9.01 
9.02 
9.03 

9.04 
9.05 
9.06 

9.07 
9.08 
9.09 

81.1801 
81.3604 
81.5409 

81.7216 
81.5)025 
82.0836 

82.2649 
82.4464 
82.6281 

3.00167 
3.00,333 
3.00500 

3.00666 
3.00832 
3.00998 

3.01164 
3.01330 
3.01496 

9.49210 
9.49737 
9.50263 

9.50789 
9.51315 
9.51840 

9.52365 
9.52890 
9.53415 

731.4.33 
733.871 
736.314 

738.763 
741.218 
743.677 

746.143 

748.613- 
751.089 

2.08085 
2.08162 
2.08239 

2.08316 

2.08393 
2.08470 

2.08546 
2.08623 
2.08699 

4.48306 
4.48472 
4.48638 

4.48803 
4.48969 
4.49134 

4.49299 
4.49-164 
4.49(529 

9.65847 
9.66204 
9.66561 

9.66918 
9.67274 
9.67630 

9.67986 
9.68.342 
9.68697 

9.10 

82.8100 

3.01662 

9.53939 

753.571 

2.08776 

4.49794 

9.69052 

9.11 
9.12 
9.13 

9.14 
9.15 
9.16 

9.17 
9.18 
9.19 

82.9921 
83.1744 
83.3569 

83.5396 
83.7225 
83.9056 

84.0889 
84.2724 
84.4561 

3.01828 
3.01993 
3.02159 

3.02324 
3.02490 
3.02655 

3.02820 
3.02985 
3.03150 

9.54463 
9.54987 
9.55510 

9.5G033 
9.565.56 
9.57079 

9.57601 
9.58123 
9.58645 

756.058 
758.551 
761.048 

763.552 
766.061 
768.575 

771.095 
773.621 
776.152 

2.08852 
2.08929 
2.09005 

2.09081 
2.09158 
2.09234 

2.09310 

2.09386 
2.09462 

4.49959 
4.50123 
4.50288 

4.50452 
4.50616 
4.50781 

4.50945 
4.51108 
4.51272 

9.69407 
9.69762 
9.70116 

9.70470 
9.70824 
9.71177 

9.71531 

9.71884 
9.722.36 

9.20 

84.6400 

3.03315 

9.59166 

778.688 

2.09538 

4.51436 

9.72589 

9.21 
9.22 
9.23 

9.24 
9.25 
9.26 

9.27 

9.28 
9.29 

84.8241 
85.0084 
85.1929 

85.3776 
85.5625 
85.7476 

85.9329 
86.1184 
86.3041 

3.03480 
3.03645 
3.03809 

3.03974 
3.04138 
3.04302 

3.04467 
3.04631 
3.04795 

9.5<)687 
9.60208 
9.60729 

9.61249 
9.61769 
9.62289 

9.62808 
9.63328 
9.(53846 

781.230 
783.777 
786.330 

788.889 
791.4.53 
794.023 

796.598 
799.179 
801.765 

2.09614 
2.09690 
2.09765 

2.09841 
2.09917 
2.09992 

2.10068 
2.10144 
2.10219 

4.51599 
4.51763 
4.51926 

4.52089 
4.-52252 
4.52415 

4.52578 
4.52740 
4.52i)03 

9.72941 
9.73293 
9.73645 

9.73996 
9.74348 
9.74699 

9.75049 
9.75400 
9.75750 

9.30 

86.4fX)0 

3.04959 

9.64365 

804.357 

2.10294 

4.53065 

9.76100 

9.31 
9.32 
9.33 

9.34 
9.35 
9.36 

9.37 
9.38 
9.39 

86.6761 
8(5.8624 
87.0489 

87.2356 
87.4225 
87.6096 

87.7969 
87.9844 
88.1721 

3.05123 
3.05287 
3.05450 

3.05614 
3.05778 
3.05941 

3.06105 
3.06268 
3.06431 

9.64883 
9.65401 
9.65919 

9.66437 
9.66954 
9.67471 

9.67988 
9.68504 
9.69020 

806.954 
809.558 
812.166 

814.781 
817.400 
820.026 

822.657 
825.294 
827.9.36 

2.10370 
2.10445 
2.10520 

2.10595 
2.10671 
2.10746 

2.10821 
2.10896 
2.10971 

4.53228 
4.5335)0 
4.53552 

4.53714 
4.53876 
4.54038 

4.54199 
4.54361 
4.54522 

9.76450 
9.76799 
9.77148 

9.77497 
9.77846 
9.78195 

9.78543 
9.78891 
9.79239 

9.40 

88..3600 

3.06594 

9.695,36 

830.584 

2.11045 

4.54684 

9.79586 

9.41 
9.42 
9.43 

9.44 
9.45 
9.46 

9.47 
9.48 
9.49 

88.5481 
88.7364 
88.9249 

89.1136 
89.3025 
89.4916 

89.6809 
89.8704 
90.0601 

3.0(>757 
3.00920 
3.07083 

3.07246 
3.07409 
3.07571 

3.07734 
3.07896 
3.08058 

9.70052 
9.705(57 
9.71082 

9.71597 
9.72111 
9.72625 

9.73139 
9.73653 
9.74166 

833.238 
835.897 
838.562 

841.232 
843.909 
846.591 

849.278 
851.971 
854.670 

2.11120 
2.11195 
2.11270 

2.11344 
2.11419 
2.11494 

2.11568 
2.11642 

2.11717 

4.54845 
4.55006 
4.55167 

4.55328 
4.55488 
4.55649 

4.55809 
4.55970 
4.56130 

9.79933 
9.80280 
9.80627 

9.80974 
9.81320 
9.81666 

9.82012 
9.82357 
9.82703 

xxvi 

Powers  and  Roots 

[11 

n 

n^ 

y/n 

VWn 

n^ 

</n 

^10  n 

</100n 

9.50 

90.2500 

3.08221 

9.74679 

857.375 

2.11791 

4.56290 

9.83048 

9.51 
9.52 
9.53 

9.54 
9.55 
9.56 

9.57 
9.58 
9.59 

90.4401 
90.6304 
90.8209 

91.0116 
91.2025 
91.3936 

91.5849 
91.7764 
91.9681 

3.08383 
3.08545 
3.08707 

3.08869 
3.09031 
3.09192 

3.09354 
3.09516 
3.09677 

9.75192 
9.75705 
9.76217 

9.76729 
9.77241 
9.77753 

9.78264 
9.78775 
9.79285 

860.085 
862.801 
865.523 

868.251 
870.984 
873.723 

876.467 
879.218 
881.974 

2.11865 
2.119i0 
2.12014 

2.12088 
2.12162 
2.12236 

2.12310 
2.12384 
2.12458 

4.56450 
4.56610 
4.56770 

4.56930 
4.57089 
4.57249 

4.57408 
4.57567 
4.57727 

9.83392 
9.83737 
9.84081 

9.84425 
9.84769 
9.85113 

9.85456 
9.85799 
9.86142 

9.60 

92.1600 

3.09839 

9.79796 

884.736 

2.12532 

4.57886 

9.86485 

9.61 
9.62 
9.63 

9.64 
9.65 
9.66 

9.67 
9.68 
9.69 

92.3521 
92.5444 
92.7369 

92.9296 
93.1225 
93.3156 

93.5089 
93.7024 
93.8961 

3.10000 
3.10161 
3.10322 

3.10483 
3.10644 
3.10805 

3.10966 
3.11127 
3.11288 

9.80306 
9.80816 
9.81326 

9.81835 
9.82344 
9.82853 

9.83362 
9.83870 
9.84378 

887.504 
890.277 
893.056 

895.841 
898.632 
901.429 

904.231 
907.039 
909.853 

2.12605 
2.12679 
2.12753 

2.12826 
2.12900 
2.12974 

2.13047 
2.13120 
2.13194 

4.58045 
4.58204 
4.58362 

4.58521 
4.58679 
4.58838 

4.58996 
4.59154 
4.59312 

9.86827 
9.87169 
9.87511 

9.87853 
9.88195 
9.88536 

9.88877 
9.89217 
9.89558 

9.70 

94.0900 

3.11448 

9.84886 

912.673 

2.13267 

4.59470 

9.89898 

9.71 

9.72 
9.73 

9.74 
9.75 
9.76 

9.77 
9.78 
9.79 

94.2841 
94.4784 
94.6729 

94.8676 
95.0625 
95.2576 

95.4529 
95.6484 
95.8441 

3.11609 
3.11769 
3.11929 

3.12090 
3.12250 
3.12410 

3.12570 
3.12730 
3.12890 

9.85393 
9.85901 
9.86408 

9.86914 

9.87421 
9.87927 

9.88433 
9.88939 
9.89444 

915.499 
918.330 
921.167 

924.010 

926.859 
929.714 

932.575 
935.441 
938.314 

2.13340 
2.1MU 
2.13487 

2.13560 
2.13(5;« 
2.13706 

2.13779 
2.13852 
2.13925 

4.59(528 
4.59786 
4.59943 

4.60101 
4.60258 
4.60416 

4.60573 
4.60730 
4.60887 

9.90238 
9.90578 
9.90918 

9.91257 
9.91596 
9.91935 

9  92274 
9.92612 
9.92950 

9.80 

96.0400 

3.13050 

9.89949 

941.192 

2.13997 

4.61044 

9.93288 

9.93626 
9.93964 
9.94301 

9.f>4638 
9.94975 
9.95311 

9.95648 
9.95984 
9.96320 

9.81 
9.82 
9.83 

9.84 
9.85 
9.86 

9.87 
9.88 
9.89 

96.2361 
96.4324 
96.6289 

96.8256 
97.0225 
97.2196 

97.4169 
97.6144 
97.8121 

3.13209 
3.13369 
3.13528 

3.13688 
3.13847 
3.14006 

3.14166 
3.14325 
3.14484 

9.90454 
9.<)0959 
9.91464 

9.91968 
9.92472 
9.92975 

9.93479 
9.93982 
9.94485 

944.076 
946.96(5 
949.862 

952.764 
955.672 
958.585 

961.505 
964.430 
967.362 

2.14070 
2.14143 
2.14216 

2.14288 
2.14361 
2.14433 

2.14506 
2.14578 
2.14651 

4.61200 
4.61357 
4.61514 

4.61670 
4.61826 
4.61983 

4.62139 
4.622{)5 
4.62451 

9.90 

98.0100 

3.14(343 

9.94987 

970.2t)9 

2.14723 

4.62607 

9.96655 

9.91 
9.92 
9.93 

9.94 
9.95 
9.96 

9.97 
9.98 
9.99 

98.2081 
98.4064 
98.6049 

98.8036 
99.0025 
99.2016 

9<).4009 
SW.6004 
i)9.8001 

3.14802 
3.14<H)0 
3.15119 

3.15278 
3.15436 
3.15595 

3.15753 
3.15911 
3.16070 

9.95490 
9.95992 
9.9641^ 

9.9(5995 
9.97497 
9.97998 

9.98499 
9.98999 
9.9^)500 

973.242 
976.191 
979.147 

982.108 
985.075 
988.048 

91^)1.027 
t)94.012 
997.003 

2.14795 

2.14867 
2.14940 

2.15012 
2.15084 
2.15156 

2.15228 
2.15300 
2.15372 

4.62762 
4.62918 
4.63073 

4.63229 

4.6;i'm 

4.63539 

4.6.3694 
4.63849 
4.(54004 

9.96991 
9.9732(5 
9.97661 

9.97996 
9.98331 
9.98665 

9.989i»9 
9.99333 
9.99667 

TABLE   III  —  IMPORTANT   NUMBERS 


A,    Units  of  Length 


English  Units 


12  inches  (in.)  =  1  foot  (ft.) 


3  feet 
6|  yards 
320  rods 


=  1  yard  (yd. ) 
=  1  rod  (rd.) 
=  1  mile  (mi.) 


English  to  Metric 

1  in.  =  2.5400  cm. 

1ft.    =30.480  cm. 
1  mi.  =  1.6093  Km. 


Metric  Units 

10  millimeters  =  1  centimeter  (cm. ) 

(mm.) 
10  centimeters  =  1  decimeter  (dm.) 
10  decimeters    =  1  meter  (m.) 
10  meters  =  1  dekameter  (Dm.) 

1000  meters  .    =  1  kilometer  (Km.) 

Metric  to  English 

1  cm.   =  0.3937  in. 

1  m.     =  39.37  in.  =  3.2808  ft. 
1  Km.  =  0.6214  mi. 


B.     Units  of  Area  or  Surface 

1  square  yard  =  9  square  feet  =  1296  square  inches 
1  acre  (A.)  =  160  square  rods  =  4840  square  yards 
1  square  mile  =  640  acres  =  102400  square  rods 

C.     Units  of  Measurement  of  Capacity 


Dry  Measure 
2  pints  (pt.)  =  1  quart  (qt.) 
8  quarts         =  1  peck  (pk.) 
4  pecks  =  1  bushel  (bu.) 


Liquid  Measure 
4  gills  (gi.)  =  1  pint  (pt.) 
2  pints  =  1  quart  (qt.) 

4  quarts       =  1  gallon  (gal.) 
1  gallon        =  231  cu.  in. 


D.    Metric  Units  to  English  Units 

1  liter  =  1000  cu.  cm.  =  61.02  cu.  in.  =  1.0667  liquid  quarts 
1  quart  =  .94636  liter  =  946.36  cu.  cm. 
1000  grams  =  1  kilogram  (Kg.)  =  2.2046  pounds  (lb.) 
1  pound  =  .453593  kilogram  =  453. 59  grams 


E.     Other  Numbers 

IT  =  ratio  of  circumference  to  diameter  of  a  circle 
=  3.14159265 
1  radian  =  angle  subtended  by  an  arc  equal  to  the  radius 

=  57°  17'  44".8  =  57°.2957795  =  18077r 
1  degree  =  0.01745329  radian,  or  7r/180  radians 
Weight  of  1  cu.  ft.  of  water  =  62.425  ]b. 
xxvii 


SYMBOLS   AND   ABBEEVIATIONS 

The  following  symbols  and  abbreviations  are  used  for  the  sake  of  brevity 
throughout  the  present  book  : 


=  equal,  or  is  equal  to 

Ax.      Axiom 

ijfc  not  equal,  or  is  not  equal  to 

Cons.  Construction,    or    by    con- 

> greater  than 

struction 

<  less  than 

Cor.     Corollary 

^  is  congruent  to 

Def.     Definition 

JL  perpendicular,  or  is  perpendic- 

Hyp.   Hypothesis,  or  by  hypoth- 

ular to 

esis 

II   parallel,  or  is  parallel  to 

Iden.    being  identical 

~  similar,  or  is  similar  to 

Prop.   Proposition 

A  angle 

rt.        right 

A  angles 

St.        straight 

A  triangle 

Th.      Theorem 

^  triangles 

Prob.   Problem 

O  parallelogram 

Fig.     Figure  or  diagram 

HJ  parallelograms 

....  and  so  on 

O  circle 

hence  or  therefore 

©  circles 

• 

^  arc 

The  signs  4- ,  — ,  x  ,  -4- ,  are  used  with  the  same  meanings  as  in  algebra. 
The  following  agreements  are  also  made  : 

a  X  b  =  a '  b  =  ab,   a-^-b  =  a/b  =  a  :  6 


xxvm 


SYLLABUS   OF   PLANE    GEOMETRY* 

INTEODUCTION 

PART   I.     DRAWING   SIMPLE   FIGURES 

PART   II.     FUNDAMENTAL   IDEAS 

PART   III.     STATEMENTS   FOR   REFERENCE 

27.   Axioms. 

1.  If  equals  are  added  to  equals^  the  sums  are  equal.  Thus,  if  a  =  6 
and  c  =  d,  then  a  -\-  c  =  b  +  d. 

2.  If  equals  are  subtracted  from  equals,  the  remainders  are  equal. 
Thus,  if  a  =  b  and  c  =  d,  then  a  —  c  =  b  —  d. 

3.  If  equals  are  multiplied  by  equals,  the  products  are  equal.  Thus, 
if  a  =  b  and  c  =  d,  then  ac  =  bd. 

4.  If  equals  are  divided  by  equals,  the  quotients  are  equal.    Thus,  if 

a  =  b  and  c  =  d,  then  -  =  -•     In  applying  this  axiom  it  is  supposed  that 
c     a 

G  and  d  are  not  equal  to  zero. 

5.  If  equals  are  added  to  unequals,  the  results  are  unequal  and  in 
the  same  order.    Thus,  if  a  =  b  and  c > d,  then  a  -\-  c^b  +  d. 

6.  If  equals  are  subtracted  from  unequals,  the  results  are  unequal 
and  in  the  same  order.     Thus,  if  a  >  &  and  c  =  d,  then  a—  c>b  —  d. 

7.  If  unequals  are  added  to  unequals  in  the  same  sense,  the  results  are 
unequal  in  the  same  order.     Thus,  if  a>b  and  c^d,  then  a  -i-  c>b  +  d. 

8.  If  unequals  are  subtracted  from  equals,  the  results  are  unequal  in 
the  opposite  order.     Thus,  if  a  =  b  and  c  >  d,  then  a  —  c<Cb  —  d. 

9.  Quantities  equal  to  the  same  quantity,  or  to  equal  quantities,  are 
equal  to  each  other.  In  other  words,  a  quantity  may  be  substituted  for 
its  equal  at  any  time  in  any  expression. 

*  This  syllabus  contains  the  principal  statements  of  fact,  with  their  article  numbers,  from 
the  Plane  Geometry  by  the  authors  of  this  book. 

xxix 


XXX  SYLLABUS  OF  PLANE  GEOMETRY 

10.  Tlie  whole  of  a  quantity  is  greater  than  any  one  of  its  parts. 

11.  The  whole  of  a  quantity  is  equal  to  the  sum  of  its  parts. 

28.   Postulates. 

1.  Only  one  straight  li?ie  can  be  drawn  joining  two  given  points. 

2.  A  straight  line  can  be  extended  indefinitely. 

3.  A  straight  line  is  the  shortest  curve  that  can  be  drawn  between  two 
points. 

4.  A  circle  can  be  described  about  any  point  as  a  center  and  with  a 
radius  of  any  length. 

6.   A  figure  can  be  moved  unaltered  to  a  new  position. 
6.  All  straight  angles  are  equal.     Hence,  also,  all  right  angles  art 
equal.,  for  a  right  angle  is  half  of  a  straight  angle. 

30.  Preliminary  Construction  Problems. 

1.  To  construct  a  triangle,  each  of  whose  sides  is  equal  to  a  given 
length.     §  3. 

2.  To  construct  a  triangle,  whose  three  sides  are,  respectively,  equal 
to  three  given  lengths.     §  4. 

3.  To  construct  a  perpendicular  to  a  given  straight  line  at  a  given 
point  in  that  line.     §  5. 

4.  To  construct  a  perpendicular  to  a  given  line  from  a  given  point  not 
on  that  line.     §  6. 

5.  To  construct,  at  a  given  point  in  a  given  line,  another  line  that 
makes  an  angle  equal  to  a  given  angle  with  the  given  line.     §  7. 

6.  To  divide  a  portion  of  a  straight  line  into  two  equal  parts.  (To 
bisect  a  line.)     §  8. 

7.  To  divide  a  given  angle  into  two  equal  parts.  (To  bisect  an  angle.) 
§9. 

31.  Preliminary  Theorems. 

1.  All  radii  of  the  same  circle  are  equal.     §  2  ;  and  §  23. 

2.  Circles  whose  radii  are  equal  can  be  placed  upon  each  other  so 
that  their  centers  and  their  circumferences  coincide  (lie  exactly  upon 
each  other) . 

3.  Equal  angles  may  be  placed  upon  each  other  so  that  their  vertices 
coincide  and  their  corresponding  sides  fall  along  the  same  straight  lines, 
This  is,  in  fact,  what  we  mean  by  equal  angles. 


SYLLABUS  OF  PLANE  GEOMETRY  xxxi 

4.  Two  straight  lines  have  at  most  one  point  in  common.     See  postu- 
late 1,  §  28. 

5.  Two  circles  have  at  most  two  points  in  common.     See  §  8. 

6.  A  straight  line  and  a  circle  may  have  at  most  two  points  in  com- 
mon. 

7.  At  a  given  point  in  a  given  line  only  one  perpendicular  can  be 
drawn  to  that  line.     (A  consequence  of  Problem  3,  §  5.) 

8.  Complements  of  the  same  angle,   or  of  equal  angles,  are  equal. 

9.  Supplements  of  the  same  angle,  or  of  equal  angles,   are  equal. 

10.  Vertical  angles  are  equal.     §  19. 

11.  If  two  adjacent  angles  have  their  exterior  sides  in  a  straight  line, 
they  are  supplementary.     §  18. 

12.  If  two  adjacent  angles  are  supplementary,  they  have  their  exterior 
sides  in  a  straight  line. 

13.  If  each  of  two  figures  can  be  placed  upon  a  third  figure  so  as  to 
coincide  with  it,  they  can  be  placed  upon  each  other  so  that  they  coincide. 

14.  Any  desired  angle  may  be  drawn,  and  any  angle  may  be  measured, 
by  the  use  of  a  protractor.  (But  the  use  of  this  instrument  is  not  per- 
mitted when  a  figure  is  to  be  constructed.     See  §  20.) 

15.  A  perpendicular  to  a  given  line  through  any  given  point  may  be 
drawn  by  means  of  a  set  square  or  a  dra^ving  triangle.  (But  the  use  of 
these  instruments  is  not  permitted  when  a  figure  is  to  be  constructed. 
See  §  20.) 

16.  The  area  of  a  rectangle  (in  terms  of  a  unit  square)  is  equal  to  the 
product  of  its  width  and  its  height,  measured  in  units  of  length  equal  to 
one  side  of  the  unit  square. 

17.  The  area  of  any  given  figure  is  greater  than  the  area  of  any  figure 
that  is  drawn  completely  within  it. 

18.  The  areas  of  two  figures  are  equal  if  they  consist  of  corresponding 
portions  that  can  be  made  to  coincide. 


CHAPTER   I 

RECTILINEAR  FIGURES 

PART  I.     TRIANGLES 

35.  Theorem  I.  If  two  triangles  have  two  sides  and  the  included 
angle  of  the  one  equal,  respectively,  to  two  sides  and  the  included 
angle  of  the  other,  the  triangles  are  congruent. 

36.  Corollary  1.  Two  right  triangles  are  congruent  if  the  two  sides  of 
the  one  are  equal,  respectively,  to  the  two  sides  of  the  other. 

37.  Theorem  II.  If  two  triangles  have  two  angles  and  the  in- 
cluded side  in  the  one  equal,  respectively,  to  two  angles  and  the 
included  side  in  the  other,  the  triangles  are  congruent. 

38.  Corollary  1.  Two  right  triangles  are  congruent  if  an  acute  angle 
and  its  adjacent  side  in  one  are  equal,  respectively,  to  'an]  acute  angle  and  its 
adjacent  side  in  the  other. 

40.  Theorem  III.  In  an  isosceles  triangle  the  angles  opposite 
the  equal  sides  are  equal. 

41 .  Corollary  1 .     If  a  triangle  is  equilateral,  it  is  also  equiangular. 

43.  Theorem  IV.  Tlie  bisector  of  the  angle  at  the  vertex  of  an 
isosceles  triangle  is  perpendicular  to  the  base  and  bisects  the  base. 

44.  Corollary  1.  In  any  isosceles  triangle  (a)  The  bisector  of  the 
angle  at  the  vertex  divides  the  triangle  into  two  congruent  right  triangles. 

(6)  The  bisector  of  the  vertical  angle  coincides  with  both  the  altitude 
and  the  median  drawn  through  the  vertex. 

(c)  The  perpendicular  bisector  of  the  base  passes  through  the  vertex, 
and  divides  the  triangle  into  two  congruent  right  triangles. 

45.  Theorem  V.  If  two  triangles  have  three  sides  of  the  one 
equal,  respectively,  to  the  three  sides  of  the  other,  they  are  congruent. 

46.  Corollary  1.  Three  sides  determine  a  triangle;  that  is,  if  the 
three  sides  are  given,  the  triangle  is  thereby  fixed. 

47.  Theorem  VI.  An  exterior  angle  of  a  triangle  is  greater  than 
either  of  the  opposite  interior  angles. 


SYLLABUS  OF  PLANE  GEOMETRY  xxxiii 

PART  II.     PARALLEL  LINES 

49.  Parallel  Postulate.  Only  one  line  can  he  drawn  through  a 
given  point  parallel  to  a  given  line. 

50.  Corollary  1.  Lines  parallel  to  the  same  line  are  parallel  to 
each  other. 

51.  Theorem  VII.  When  two  lines  are  cut  by  a  transversal,  if 
the  alternate  interior  angles  are  equal,  the  two  lines  are  parallel. 

52.  Corollary  1.     Lines  perpendicular  to  the  same  line  are  parallel. 
54.    Theorem  VIII.     If  two  parallel  lines  are  cut  by  a  transversal, 

the  alternate  interior  angles  are  equal. 

56.  Theorem  IX.  If  two  lines  are  cut  by  a  transversal  and  the  cor- 
responding angles  are  equals  the  lines  are  parallel. 

57.  Corollary  1.  If  two  lines  are  cut  by  a  transversal  and  the  two 
interior  angles  on  the  same  side  of  the  transversal  are  supplementary^  the 
lines  are  parallel. 

58.  Corollary  2.  From  a  given  point  only  one  perpendicular  can  be 
drawn  to  a  given  line. 

59.  Theorem  X.  {Converse  of  Theorem  IX.)  If  two  parallel 
lines  are  cut  by  a  transversal.,  the  corresponding  angles  are  equal. 

60.  Corollary  1 .  If  a  line  is  perpendicular  to  one  of  two  parallels^ 
it  is  perpendicular  to  the  other  also. 

61.  Problem  1.  To  construct  a  line  parallel  to  a  given  line  and 
passing  through  a  given  point. 

PART  III.     ANGLES  AND  TRIANGLES 

62.  Theorem  XI.  The  sum  of  three  angles  of  any  triangle  is 
^qual  to  two  right  angles,  or  180°. 

63.  Corollary  1 .  The  sum  of  the  two  acute  angles  of  any  right  triangle 
is  one  right  angle,  or  90°. 

64.  Corollary  2.  An  exterior  angle  of  any  triangle  is  equal  to  the  sum 
of  its  opposite  interior  angles. 

65.  Corollary  3.  Each  angle  of  an  equilateral  triangle  is  equal  to 
60°. 


xxxiv  SYLLABUS  OF  PLANE  GEOMETRY 

66.  Corollary  4.  If  two  angles  of  one  triangle  are  equals  respectively^ 
to  two  angles  of  another  triangle^  then  the  third  angles  are  likewise  equal, 

67.  Theorem  XII.  Tico  angles  whose  sides  are  respectively 
parallel  are  either  equal  or  supplementary. 

68.  Theorem  XIII.  Two  angles  whose  sides  are  respectively 
perpendicular  to  each  other  are  either  equal  or  supplementary. 

69.  Theorem  XIV.  Two  right  triangles  are  congruent  if  the 
hypotenuse  and  an  acute  angle  of  the  one  are  equal,  respectively,  to  the 
hypotenuse  and  an  acute  angle  of  the  other. 

70.  Theorem  XV-  Two  right  triangles  are  congruent  if  the 
hypotenuse  and  a  side  of  the  one  are  equal,  respectively,  to  the  hypote- 
nuse and  a  side  of  the  other. 

71.  Corollary  1.  If  two  oblique  lines  of  equal  length  are  drawn 
from  a  point  G  in  a  perpendicular  CD  to  a  line  AB,  they  cut  off  equal 
distances  from  the  foot  of  the  perpendicular,  and  conversely. 

72.  Theorem  XVI.  {Converse  of  Theorem  III.)  If  two  angles 
of  a  triangle  are  equal,  the  sides  opposite  are  equal,  and  the  triangle  is 
isosceles. 

73.  Corollary  1.     An  equiangular  triangle  is  also  equilateral. 

74.  Corollary  2.  If  two  oblique  lines  are  draionfrom  a  point  C  in 
a  perpendicular  CD  to  a  line  AB,  so  as  to  make  equal  angles  with  AB, 
they  are  equal,  and  conversely. 

75 .  Theorem  XVII .  If  two  angles  of  a  triangle  are  unequal,  the  sides 
opposite  them  are  unequal  and  the  greater  side  is  opposite  the  greater  angle. 

76.  Corollary  1.  If  two  triangles  have  two  sides  of  the  one  equal  to 
two  sides  of  the  other,  but  the  included  angle  of  the  first  greater  than  the 
included  angle  of  the  second,  then  the  third  side  of  the  first  is  greater  than 
the  third  side  of  the  second. 

77.  Corollary  2.  If  two  oblique  lines  are  drawn  from  a  point  G  in 
a  perpendicular  CD  to  a  line  AB,  and  if  the  base  angles  at  A  and  B  are 
unequal,  the  oblique  line  opposite  the  greater  base  angle  is  the  greater ; 
in  particular,  the  perpendicular  CD  is  itself  the  shortest  line  from  C  to 
any  point  of  AB. 


SYLLABUS  OF  PLANE  GEOMETRY  XXXV 

78.  Theorem  XVIII.  If  two  sides  of  a  triangle  are  unequal,  the 
angles  opposite  them  are  unequal  and  the  greater  angle  is  opposite  the 
greater  side. 

79.  Corollary  1.  If  two  triangles  have  two  sides  of  the  one  equal  to 
two  sides  of  the  other,  hut  the  third  side  of  the  first  greater  than  the  third 
side  of  the  second,  then  the  included  angle  of  the  first  is  greater  than  the 
included  angle  of  the  second. 

80.  Corollary  2.  If  from  a  point  C  in  a  perpendicular  CD  to  a 
line  AB  uneqiial  oblique  lines  are  drawn  to  the  base  AB,  the  longer  of 
the  oblique  lines  is  opposite  the  larger  of  the  two  base  angles. 

PART  IV.     QUADRILATERALS 

82.  Theorem  XIX.  Either  diagonal  of  a  parallelogram  divides  it 
into  two  congruent  triangles. 

83.  Corollary  1.  Any  side  of  a  parallelogram  is  equal  to  the  side 
opposite  it. 

84.  Corollary  2.  The  segments  of  parallel  lines  included  between 
parallel  lines  are  equal. 

85.  Theorem  XX.  If  a  quadrilateral  has  both  pairs  of  opposite 
sides  equal,  it  is  a  parallelogram. 

86.  Theorem  XXI.     If  a  quadrilateral  has  one  pair  of  sides  equal 

and  parallel,  it  is  a  par'allelogram. 

87.  Theorem  XXII.  The  diagonals  of  a  parallelogram  bisect  each 
other. 

88.  Theorem  XXIII.  Two  parallelograms  are  congruent  if  two 
sides  and  the  included  angle  of  the  one  are  equal,  respectively,  to  two 
sides  and  the  included  angle  of  the  other. 

89.  Theorem  XXIV.  The  line  joining  the  middle  points  of  the 
two  sides  of  a  triangle  is  parallel  to  the  base  and  equal  to  half  the 
base. 

90.  Corollary  1.  {Converse  of  %  89.)  The  line  drawn  through  the 
middle  point  of  one  side  of  a  triangle  parallel  to  the  base  bisects  the  other 
side. 


xxxvi  SYLLABUS  OF  PLANE  GEOMETRY 

91 .  Theorem  XXV.  If  three  parallel  lines  cut  off  two  equal  por- 
tions of  one  transversal,  they  cut  off  two  equal  portions  of  any  other 
transversal, 

92.  Corollary  1.  If  a  series  of  parallel  lines  cut  off  equal  portions 
of  one  transversal^  they  cut  off  equal  portions  of  any  other  transversal. 

93.  Corollary  2.  If  three  parallel  lines  cut  off  two  portions  of  one 
transversal,  one  of  which  is  double  the  other,  they  cut  off  two  portions  of 
any  other  transversal,  one  of  which  is  double  the  other. 

94.  Corollary  3.  If  three  parallel  lines  cut  off  two  2yortions  of  one 
transversal,  one  of  which  is  n  times  the  other,  they  cut  off  two  portions  of 
any  other  transversal,  one  of  which  is  n  times  the  other. 

PART   V.     POLYGONS 

97.  Theorem  XXVI.  TTie  sum  of  the  interior  angles  of  a  polygon 
is  two  right  angles  taken  as  many  times  as  the  figure  has  sides,  less  two. 

98.  Theorem  XXVII.  The  sum  of  the  exterior  angles  of  a  polygon 
formed  by  producing  the  sides  in  succession  is  equal  to  four  right  angles. 

PART   VI.     THE   LOCUS   OF  A   POINT 

100.  Theorem  XXVIII.  The  locus  of  all  points  equidistant  from 
the  extremities  of  a  line  is  the  perpendicular  bisector  of  that  line. 

101.  Theorem  XXIX.  The  bisector  of  an  angle  is  the  locus  of 
all  points  equidistant  from  its  sides. 

102.  Supplementary  Propositions  on  Altitudes,  Medians,  etc. 

Theorem  XXX.  The  perpendiculars  erected  at  the  middle  points  of 
the  sides  of  a  ti'iangle  meet  in  a  point. 

Theorem  XXXI.  The  bisectors  of  the  angles  of  a  triangle  meet  in  a 
point. 

Theorem  XXXII.      The  altitudes  of  a  triangle  meet  in  a  point. 

Theorem  XXXIII.  The  medians  of  a  triangle  meet  in  a  point 
which  is  two  thirds  of  the  distance  from  any  vertex  to  the  middle  point  of 
the  opposite  side. 


SYLLABUS  OF  PLANE  GEOMETRY         xxxvii 

CHAPTER   II 
THE   CIRCLE 

PART   L     CHORDS.    ARCS.    CENTRAL  ANGLES 

104.   Postulates. 

(1)  As  a  central  angle  increases,  its  intercepted  arc  increases,  and 
vice  versa  ;  and  as  a  central  angle  decreases,  its  intercepted  arc  decreases, 
and  vice  versa. 

(2)  l7i  the  same  circle  (or  equal  circles},  equal  central  angles  intercept 
equal  arcs  ;  and  equal  arcs  subtend  equal  central  angles. 

106.  Theorem  I.  In  the  same  circle  (or  in  equal  circles),  equal 
arcs  subtend  equal  chords. 

107.  Theorem  II.  (Converse  of  §  106.)  In  the  same  circles  (or 
in  equal  circles),  equal  chords  subtend  equal  arcs. 

108.  Theorem  III.  A  diameter  perpendicular  to  a  chord  bisects 
the  chord  and  the  arc  subtended  by  it, 

109.  Theorem  IV.  In  the  same  circle  (or  in  equal  circles) j 
equal  chords  are  equally  distant  from  the  center,  and,  conversely, 
chords  that  are  equally  distant  from  the  center  are  equal. 

110.  Theorem  V.  In  the  same  circle  (or  in  equal  circles),  if  two 
unequal  chords  are  drawn,  the  longer  one  is  nearer  the  center. 

111.  Corollary  1.  (Converse  of  §  110.)  In  the  same  circle  (or  in 
equal  circles),  if  two  chords  are  unequally  distant  from  the  center,  the  more 
remote  is  the  less. 

PART  XL     TANGENTS  AND  SECANTS 

115.  Theorem  VI.  A  line  perpendicular  to  a  radius  at  its  ex- 
tremity is  tangent  to  the  circle. 

116.  Corollary  1.  A  tangent  to  a  circle  is  perpendicular  to  the 
radius  drawn  to  the  point  of  contact. 

117.  Corollary  2.  A  perpendicular  to  a  tangent  at  its  point  of  con- 
tact passes  through  the  center  of  the  circle. 

118.  Theorem  VII.  Two  tangents  draion  to  a  circle  from  a  point 
outside  are  of  equal  length. 


xxxviii       SYLLABUS  OF  PLANE  GEOMETRY 

119.  Theorem  VIII.  Two  parallel  lines  intercept  equal  arcs 
on  a  circle. 

120.  Theorem  IX.  Through  three  given  points  not  all  on  the 
same  straight  line,  one  and  only  one  circle  can  he  drawn. 

121.  Corollary  1.  A  circle  may  he  drawn  to  circumscrihe  any 
triangle. 

122.  Corollary  2.  The  perpendicular  bisectors  of  the  sides  of  a 
triangle  meet  in  a  point. 

123.  Corollary  3.     A  circle  may  he  completed  if  any  arc  of  it  is  given. 

124.  Theorem  X.     A  cii^cle  may  he  inscribed  in  any  triangle. 

126.  Corollary  1.  A  circle  drawn  from  any  point  on  the  hisector 
of  an  angle,  with  a  radius  equal  to  the  distance  from  that  point  to  one 
side,  is  tangent  to  hoth  sides  of  the  angle. 

PART  III.     MEASUREMENT   OF  ANGLES 

130.  Theorem  XI.  In  the  same  circle  (or  in  equal  circles),  two 
central  angles  have  the  same  ratio  as  their  intercepted  arcs. 

Note  1.  We  may  assume  as  a  postulate  that  if  two  geometric  ratios 
are  equal  whenever  their  terms  are  commensurable,  they  are  equal  also 
when  their  terms  are  incommensurable. 

Note  2.     A  central  angle  is  measured  by  its  intercepted  arc. 

132.  Theorem  XII.  An  inscribed  angle  is  measured  by  one  half 
of  its  intercepted  arc. 

133.  Corollary  1.  Any  angle  inscribed  in  a  semicircle  is  a  right 
angle. 

134.  Corollary  2.  Aiiy  angle  inscribed  in  a  segment  greater  than 
a  semicircle  is  acute,  while  any  angle  inscribed  in  a  segment  less  than  a 
semicircle  is  obtuse. 

135 .  Corollary  3.    All  angles  inscribed  in  the  same  segment  are  equal. 

136.  Theorem  XIII.  An  angle  formed  hy  two  chords  inter- 
secting within  a  circle  is  measured  hy  one  half  of  the  sum  of  the 
intercepted  arcs. 

137.  Theorem  XIV.  An  angle  formed  by  a  tangent  and  a  chord 
drawn  through  the  point  oftangency  is  measured  by  one  half  of  the  inter- 
cepted arc. 


SYLLABUS  OF  PLANE  GEOMETRY  xxxix 

138.  Theorem  XV.  An  angle  formed  by  two  secants^  or  by  a  tan- 
gent and  a  secant,  or  by  two  tangents  that  meet  outside  a  circle  is  meas- 
ured by  one  half  the  difference  of  the  intercepted  arcs. 

PART  IV.  CONSTRUCTION  PROBLEMS 

139.  Problem  1.  Through  a  given  point  to  draw  a  tangent  to 
a  circle. 

140.  Problem  2.      To  circumscribe  a  circle  about  a  given  triangle. 

141.  Problem  3.      To  inscribe  a  circle  in  a  given  triangle. 

142.  Problem  4.  On  a  given  straight  line  to  construct  a  segment  of 
a  circle  that  shall  contain  a  given  angle. 

CHAPTER   III 

PROPORTION.    SIMILARITY 

PART  I.     GENERAL  THEOREMS  ON  PROPORTION 

144.    General  Theorems  on  Proportion. 

Theorem  A.  In  any  proportion,  the  product  of  the  extremes  is  equal 
to  the  product  of  the  means. 

Corollary  1 .  If  the  two  antecedents  of  a  proportion  are  equal,  the 
consequents  are  also  equal. 

Theorem  B.  If  the  product  of  two  numbers  is  equal  to  the  product  of 
two  other  numbers,  either  pair  may  be  made  the  means  of  a  proportion  in 
which  the  other  two  are  taken  as  the  extremes. 

Theorem  C.  if  four  quantities  are  in  proportion,  they  are  in  propor- 
tion by  inversion. 

Theorem  D.  If  four  quantities  are  in  proportion,  they  are  in  propor- 
tion by  alternation. 

Theorem  E.  If  four  quantities  are  in  proportion,  they  are  in  propor- 
tion by  composition. 

Theorem  F.  If  four  quantities  are  in  proportion,  they  are  in  propor- 
tion by  division. 


xl  SYLLABUS  OF  PLANE  GEOMETRY 

Theorem  G.  If  four  quantities  are  in  proportion^  they  are  in  propor- 
tion by  composition  and  division. 

Theorem  H.  Li  a  series  of  equal  ratios  the  sum  of  the  antecedents 
is  to  the  sum  of  the  consequents  as  any  antecedent  is  to  its  consequent. 

PAET   II.     PROPORTIONAL    LINE-SEGMENTS 

145.  Theorem  I.  A  line  parallel  to  the  base  of  a  triangle  divides 
the  other  sides  proportionally. 

146.  Corollary  1.  If  a  line  is  drawn  parallel  to  the  base  of  a  tri- 
angle, either  side  is  to  one  of  its  segments  as  the  other  side  is  to  its 
corresponding  segment. 

147.  Theorem  II.  (Converse  of  Theorem  I.)  If  a  line  divides 
tioo  sides  of  a  triangle  proportionally ,  it  is  parallel  to  the  third  side. 

148.  Corollary  1.  If  a  line  cuts  two  sides  of  a  triangle  in  such  a 
way  that  either  side  is  to  one  of  its  segments  as  the  other  side  is  to  its 
corresponding  segment,  then  the  line  is  parallel  to  the  third  side. 

149.  Theorem  III.  The  bisector  of  an  angle  of  a  triangle  divides 
the  opposite  side  into  segments  which  are  proportional  to  the  sides  of  the 
angle. 

150.  Theorem  IV.  If  a  series  of  parallels  be  cut  by  two  lines, 
the  corresponding  segments  are  proportional. 

151.  Problem  1.  To  divide  a  given  line  into  parts  propor- 
tional to  any  number  of  given  lines. 

153.  Problem  2.  To  find  the  fourth  prop^ortional  to  three 
given  lines. 

PART  III.     SIMILAR  TRIANGLES   AND   POLYGONS 

155.  Theorem  V.  If  two  triangles  are  mutually  equiangular^ 
they  are  similar. 

156.  Corollary  1.  Two  triangles  are  similar  if  two  angles  of  the 
one  are  equal  respectively  to  two  angles  of  the  other. 

167.  Corollary  2.  Tv^o  right  triangles  are  similar  if  an  acute  angle 
of  the  one  is  equal  to  an  acute  angle  of  the  other. 

158.  Theorem  VI.  {Converse  of  Theorem  V.)  If  two  tri- 
angles are  similar,  they  are  mutually  equiangular. 


SYLLABUS  OF  PLANE  GEOMETRY  xli 

159.  Theorem  VII.  Two  triangles  are  similar  if  an  angle  of 
the  one  equals  an  angle  of  the  other  and  the  including  sides  are 
proportional. 

161.  Theorem  VIII.  If,  in  any  right  triangle,  a  perpendicular  is 
drawn  from  the  vertex  of  the  right  angle  to  the  hypotenuse,  the  two  right 
triangles  thus  formed  are  similar  to  each  other  and  to  the  given  triangle. 

162.  Corollary  1.  In  any  right  triangle  the  perpendicular  from  the 
vertex  of  the  right  angle  to  the  hypotenuse  is  the  mean  proportional  be- 
tween the  segments  of  the  hypotenuse. 

163.  Corollary  2.  If  in  any  right  triangle,  a  perpendicular  is 
draion  from  the  vertex  of  the  right  angle  to  the  hypotenuse,  each  side  of 
the  right  triangle  is  the  mean  proportional  between  the  hypotenuse  and 
the  segment  adjacent  to  that  side. 

164.  Problem  3.     To  find  the  mean  proportional  between  two 


166.  Theorem  IX.  Regular  polygons  of  the  same  number  of 
sides  are  similar. 

167.  Theorem  X.  The  perimeters  of  two  similar  polygons  are 
to  each  other  in  the  same  ratio  as  any  two  correspondiyig  sides. 

PART  IV.  PROPORTIONAL  PROPERTIES  OF  CHORDS, 
SECANTS,  AND  TANGENTS 

168.  Theorem  XL  If  two  chords  intersect  within  a  circle,  the 
product  of  the  segments  of  the  one  is  equal  to  the  prodiict  of  the 
segments  of  the  other. 

169.  Theorem  XII.  If  from  a  point  without  a  circle  a  secant 
and  a  tangent  are  drawn,  the  tangent  is  the  mean  proportional 
between  the  entire  secant  and  its  exterior  segment. 

170.  Theorem  XIII.  If  from  a  fixed  point  without  a  circle 
any  two  secants  are  drawn,  the  product  of  one  secant  and  its 
external  segment  is  equal  to  the  product  of  the  other  secant  and  its 
external  segment. 

172.  Problem  4.  To  divide  a  given  line  segment  in  extreme 
arid  mean  ratio. 


xlii  SYLLABUS  OF  PLANE  GEOMETRY 

PART  V.     SIMILAR  RIGHT  TRIANGLES.     TRIGONOMETRIC 

RATIOS 

174.  Theorem  XIV.  If  two  right  triangles  have  one  acute  angle 
of  one  equal  to  one  acute  angle  of  the  other,  their  corresponding  sides 
are  in  the  same  ratios. 

175.  Corollary  1.  If  an  acute  angle  of  a  right  triangle  is  known, 
the  ratios  of  the  sides  are  all  determined. 

176.  Corollary  2.  If  the  ratio  of  any  pair  of  sides  of  a  right  tri- 
angle is  given,  the  acute  angles  are  determined. 

179.  Theorem  XV.  Corresponding  altitudes  divide  any  two 
similar  triangles  into  two  corresponding  pairs  of  similar  right 
triangles. 

180.  Corollary  1.  Any  two  similar  polygons  may  he  subdivided  into 
corresponding  pairs  of  similar  right  triangles. 

CHAPTER   IV 
AREAS   OF  POLYGONS.    PYTHAGOREAN  THEOREM 

181.  Area  of  a  Rectangle.  The  area  of  a  rectangle  is  equal  to  the 
product  of  its  base  by  its  height. 

182.  Corollary  1.  The  area  of  a  square  is  equal  to  the  square  of  its 
side. 

183.  Corollary  2.  The  areas  of  two  rectangles  are  to  each  other  as 
the  products  of  their  bases  and  altitudes. 

184.  Corollary  3.  Two  rectangles  that  have  equal  altitudes  are  to 
each  other  as  their  bases;  two  rectangles  that  have  equal  bases  are  to  each 
other  as  their  altitudes. 

186.  Theorem  I.  The  area  of  a  parallelogram  is  equal  to  the 
product  of  its  base  by  its  altitude. 

187.  Corollary  1.  (a)  Two  parallelograms  are  to  each  other  as  the 
products  of  their  bases  and  altitudes. 

(6)  Two  parallelograms  that  have  equal  bases  and  equal  altitudes  are 
equal  in  area. 


SYLLABUS  OF  PLANE  GEOMETRY  xliii 

188.  Corollary  2.  Two  parallelograms  that  have  equal  altitudes 
are  to  each  other  as  their  bases  ;  two  parallelograms  that  have  equal  bases 
are  to  each  other  as  their  altitudes. 

189.  Theorem  II.  The  area  af  a  triangle  is  equal  to  one  half 
the  product  of  its  base  by  its  altitude. 

190.  Corollary  1.  (a)  Two  triangles  are  to  each  other  as  the 
products  of  their  bases  and  altitudes. 

(b)  Tioo  triangles  that  have  equal  bases  are  to  each  other  as  their 
altitudes. 

(c)  Tvjo  triangles  that  have  equal  altitudes  are  to  each  other  as  their 
bases. 

(d)  Tioo  triangles  that  have  equal  bases  and  equal  altitudes  are  equal 
in  area. 

191.  Theorem  III.  The  area  of  a  trapezoid  is  equal  to  the 
product  of  its  altitude  and  one  half  the  sum  of  its  bases. 

192.  Corollary  1.  The  area  of  a  trapezoid  is  equal  to  the  product 
of  its  altitude  and  the  line  joining  the  mid-points  of  the  non-parallel  sides. 

The  area  of  a  trapezoid  is  equal  to  the  product  of  its  altitude  and  its 
median. 

193.  Theorem  IV.  Two  triayigles  that  have  an  acute  angle  of 
the  one  equal  to  an  acute  angle  of  the  other  are  to  each  other  as  the 
products  of  the  sides  including  the  equal  angles. 

194.  Theorem  V.  Similar  triangles  are  to  each  other  as  the 
squares  of  any  two  corresponding  sides. 

195.  Corollary  1.  The  areas  of  two  similar  polygons  are  to  each 
other  as  the  squares  of  any  two  corresponding  sides. 

TJie  areas  of  two  similar  polygons  are  to  each  other  as  the  squares  of 
any  two  corresponding  lines. 

196.  Theorem  VI.  The  Pythagorean  Theorem.  The  square  on 
the  hypotenuse  of  a  right  triangle  is  equivalent  to  the  sum  of  the 
squares  on  the  two  sides. 

197.  Corollary  1.  The  square  on  either  side  of  a  right  triangle  is 
equivalent  to  the  square  on  the  hypotenuse  diminished  by  the  square  on 
*he  other  side. 


xliv  SYLLABUS  OF  PLANE  GEOMETRY 

199.  Theorem  VII.  In  any  triangle  the  square  on  the  side 
opposite  an  acute  angle  is  equal  to  the  sum  of  the  squares  on  the 
other  two  sides  diminished  by  twice  the  product  of  one  of  those 
sides  and  the  projection  of  the  other  upon  it. 

200.  Theorem  VIII.  In  any  obtuse  triangle  the  square  on  the 
side  opposite  the  obtuse  angle  is  equal  to  the  sum,  of  the  squares  on 
the  other  sides  increased  by  twice  the  product  of  one  of  those  sides 
and  the  projection  of  the  other  upon  it. 

201.  Problem  1.  To  construct  a  square  whose  area  shall  be  equal 
to  the  sum  of  the  areas  of  two  given  squares. 

202.  Problem  2.  To  construct  a  triangle  whose  area  shall  he  equal 
to  that  of  a  given  polygon. 

CHAPTER  V 
REGULAR  POLYGONS  AND   CIRCLES 

204.  Theorem  I.  If  a  circle  is  divided  into  a  number  of  equal 
arcs  : 

(a)  the  chords  joining  the  points  of  division  form  a  regidar 
inscribed  polygoyi ; 

(6)  tangents  drawn  at  the  points  of  division  form  a  regular 
circumscribed  polygon. 

205.  Theorem  II.  (a)  A  circle  may  be  circumscribed  about 
any  regular  polygon  ;  (b)  a  circle  may  also  be  inscribed  in  it. 

207.  Theorem  III.  Tlie  area  of  a  regular  polygon  is  equal  to 
half  the  product  of  its  apothem  and  perimeter. 

210.  Areas  and  Lengths  of  Circles. 

If  the  number  of  sides  of  the  regular  inscribed  and  regular  circum- 
scribed polygons  is  repeatedly  doubled : 

(a)  their  areas  approach  the  area  of  the  circle  as  a  common  limit ; 

(b)  their  perimeters  approach  the  length  of  the  circumference  of  the 
circle  as  a  common  limit. 

211.  Theorem  IV.  The  circumferences  of  two  circles  are  to  each 
other  as  their  radii. 


SYLLABUS  OF  PLANE  GEOMETRY  xlv 

212.  Corollary  1.      The  ratio  of  a  circumference  to  its  diameter  is 

the  same  for  all  circles. 

213.  The  Number  it.  The  number  obtained  by  dividing  the  circum- 
ference of  any  circle  by  its  diameter  is  denoted  by  the  Greek  letter  tt. 

214.  Corollary  2.  In  any  circle  c  =  ird,  or  c  =  %  irr,  where  r  is 
the  radius,  d  the  diameter,  and  c  the  length  of  the  circumference. 

215.  Theorem  V.  The  area  of  a  circle  is  equal  to  one  half  the 
product  of  its  radius  and  its  circumference. 

216.  Corollary  1.  The  area  of  a  circle  is  equal  to  it  times  the 
square  of  its  radius,  that  is,  A  =  Trr^. 

217.  Corollary  2.  The  areas  of  two  circles  are  to  each  other  as  the 
squares  of  their  radii. 

220.  Problem  1.  Given  the  side  and  radius  of  a  regular  inscribed 
polygon,  to  find  the  side  of  a  regular  inscribed  polygon  of  double  the 
number  of  sides. 

221.  Corollary  1.  If  r  =  1,  and  s  =  the  side  of  the  inscribed 
polygon,  the  side  AC  of  the  regular  inscribed  polygon  of  double  the  num- 
ber of  sides  is,  

222.  Problem  2.      To  compute  approximately  the  value  of  tt. 

TT  =  M§^  (approximately)  =  3.14159  (usually  written  3.1416,  or  3}). 

223.  Problem  3.      To  inscribe  a  square  in  a  given  circle. 

224.  Problem  4.      To  inscribe  a  regular  hexagon  in  a  given  circle. 

225.  Problem  5.      To  inscribe  a  regular  decagon  in  a  given  circle. 


APPENDIX  TO   PLANE   GEOMETRY 
MAXIMA  AND   MINIMA 

226.  Of  all  chords  through  P,  the  diameter  PQ  is  the  maximum 
(greatest) . 

Of  all  regular  polygons  inscribed  in  a  circle,  the  equilateral  triangle 
has  the  minimum  (least)  area,  or  simply,  is  the  minimum. 

Of  all  the  straight  lines  that  can  be  drawn  from  a  fixed  point  to  a  given 
line,  the  perpendicular  is  the  minimum. 


xlvi  SYLLABUS  OF  PLANE  GEOMETRY 

227.  Theorem  I.  Of  all  triangles  that  have  the  same  two  given 
sides,  that  in  which  these  sides  include  a  right  angle  is  the  maximum. 

229.  Theorem  II.  Of  all  isoperimetric  triangles  having  the  same 
base,  the  isosceles  is  the  maximum. 

230.  Corollary  1.  Of  all  isoperimetric  triangles,  the  equilateral  is 
the  maximum. 

231.  Theorem  III.  Of  all  isoperimetric  polygons  having  the  same 
number  of  sides,  the  maximum  is  equilateral. 

232.  Theorem  IV.  Of  all  polygons  with  sides  all  given  but  one,  the 
maximum  (in  area)  can  be  inscribed  in  a  semicircle  having  the  unde- 
termined side  for  its  diameter. 

233.  Theorem  V.  Of  all  polygons  icith  the  same  given  sides,  that 
vjhich  can  be  inscribed  in  a  circle  is  the  maximum. 

234.  Corollary  1.  Of  all  isoperimetric  polygons  of  a  given  number 
of  sides,  the  maximum  is  regular. 

235.  Theorem  VI.  Of  two  isoperimetric  regular  polygons,  the  one 
having  the  greater  number  of  sides  has  the  greater  area. 

236.  Corollary  1.  The  circle  is  the  maximum  of  all  isoperimetric 
plane  closed  figures. 

237.  Theorem  VII.  Of  all  regular  polygons  of  the  same  area,  that 
which  has  the  greatest  number  of  sides  has  the  minimum  perimeter. 

238.  Corollary  1.  Of  all  plane  closed  figures  that  are  equal  in  area, 
the  circle  has  the  minimum  perimeter. 


INDEX 

[Numbers  refer  always  to  pages.] 


Altitude,  of  prism,  239  ;  of  pyramid, 
253  ;  of  frustum  of  pyramid,  254  ; 
of  cylinder,  263  ;  of  cone,  268  ;  of 
zone,  311. 

Angle,  convex  polyhedral,  232 ;  di- 
hedral, 228;  of  spherical  poly- 
gon, 297  ;  plane,  228 ;  polyhedral, 
232 ;  spherical,  296 ;  symmetric 
trihedral,  308;  trihedral,  232; 
units  of,  202. 

Area,  lateral,  see  Lateral  area ;  of 
sphere,  310;  of  zone,  311;  of 
lune,  312,  313;  of  spherical 
triangle,  317. 

Axis,  of  cone,  268 ;  of  circle  of  a 
sphere,  288. 

Base,    of    prism,    238;    of    cylinder, 

263. 
Birectangular  triangle,  304. 

Cavalieri's  theorem,  279. 

Center,  of  sphere,  284. 

Circumscribed  sphere,  294. 

Cone,  268  ;  altitude  of,  268 ;  axis  of, 
268  ;  circular,  268  ;  right  circular, 
269;  slant  height  of,  269;  frus- 
tum of,  269 ;  lateral  surface  of, 
269;  of  revolution,  269;  pris- 
moid  formula  for,  280  ;  volume  of, 
272. 

Congruent  polyhedral  angles,  232. 

Congruent  solids,  242. 

Conical  surface,  288. 

Cube,  245  ;   volume  of,  249. 

Cylinder,  263;  circular,  263;  right, 
263  ;  right  circular,  263  ;  of  revo- 
lution,  264  ;  prismoid  formula  for. 


280  ;  lateral  area  of,  265 ;  volume 
of,  266. 
Cylindrical  surface,  263. 

Diagonal,  of  polyhedron,  238;  of 
spherical  polygon,  297. 

Diameter,  of  sphere,  284. 

Dihedral  angle,  228;  measure  of, 
228 ;  plane  angle  of,  228 ;  equal, 
228 ;    faces  of,  228 ;    edge  of,  228. 

Dimensions,  Geometry  of  three, 
215 ;  of  a  rectangular  parallele- 
piped, 248. 

Directrix,  of  cylindrical  surface,  263. 

Distance  on  a  sphere,  288. 

Dodecahedron,  238. 

Edge,  of  dihedral  angle,  228;  of 
polyhedral  angle,  232;  of  poly- 
hedron, 238. 

Element  of  cylindrical  surface,  263. 

Equivalent  solids,  245. 

Face,  of  dihedral  angle,  228;  of 
polyhedral  angle,  232 ;  of  poly- 
hedron, 238. 

Face  angle,  232. 

Frustum,  of  pyramid,  254 ;  of  cone, 
269;  lateral  area  of,  255,  271; 
volume  of,  261,  262,  272. 

Generatrix,    of    cylindrical    surface, 

263. 
Great  circle  of  sphere,  288. 

Hexahedron,  238. 

Icosahedron,  238. 
Inscribed  sphere,  294. 
Isosceles  spherical  triangle,  297, 


xlvii 


xlviii 


INDEX 


Lateral  area,  of  prism,  238,  241 ;  of 
regular  pyramid,  255 ;  of  frustum 
of  regular  pyramid,  255 ;  of  any 
cylinder,  265 ;  of  right  circular 
cylinder,  265 ;  of  any  circular 
cylinder,  265 ;  of  right  circular 
cone,  271 ;  of  frustum  of  right 
circular  cone,  271. 

Lateral  face,  of  prism,  238  ;  of  pyra- 
mid, 253;  of  frustum  of  regular 
pyramid,  254. 

Lune,  312  ;  area  of,  312,  313. 

Oblique  prism,  239. 
Octahedron,  238. 

Parallel  lines,  219. 

Parallel  planes,  219. 

Parallelepiped,  245  ;  right,  •  245  ; 
rectangular,  245 ;  oblique,  245 ; 
volume  of,  250. 

Perpendicular  planes,  226. 

Plane,  215 ;  how  determined,  215 ; 
line  perpendicular  to,  219 ;  per- 
pendicular planes,  226 ;  parallel 
planes,  219  ;  projecting,  237. 

Point  of  tangency,  285. 

Polar  distance,  291. 

Polar  triangle,  300. 

Pole  of  circle  of  sphere,  288. 

Polygon,  spherical,  297. 

Polyhedral  angle,  232  ;  convex,  232  ; 
congruent,  232;  faces  of,  232; 
edges  of,  232 ;  vertex  of,  232. 

Polyhedron,  238  ;  edges  of,  238 ;  ver- 
tices of,  238 ;  faces  of,  238 ;  diag- 
onal of,  238  ;  similar,  276 ;  regular, 
276. 

Postulate,  on  the  plane,  216 ;  on  the 
cylinder,  264, 

Prism,  238;  bases  of,  238;  lateral 
faces  of,  238;  lateral  area  of, 
238 ;  altitude  of,  239  ;  right,  239  ; 
oblique,  239;  regular,  239;  right 
section  of,  239 ;  congruent,  242 ; 
truncated,  243 ;  volume  of,  251 ; 
inscribed,  264 ;  prismoid  formula 
for,  280  ;  lateral  area  of,  241. 

Prismoid,  281. 

Prismoid  formula,  280. 


Projecting  plane,  237. 

Projection,  237. 

Pyramid,  253 ;  lateral  faces  of,  253 

vertex   of,  253;    altitude  of,  253 

regular,  253 ;   slant  height  of,  253 

truncated,  254 ;    frustum  of,  254 
inscribed,  270 ;    prismoid  formula 

for,    280;     lateral    area    of,  255; 
volume  of,  261. 

Quadrant,  291. 

Radius,  of  a  sphere,  284. 
Regular  polyhedron,  276. 
Regular  prism,  239. 
Regular  pyramid,  253. 
Regular  soHds,  276. 
Right  prism,  239. 

Right    section,    of    prism,    239 ;     of 
cylinder,  263. 

Similar  polyhedrons,  276. 
Similar  tetrahedrons,  275. 
Slant   height,   of   pyramid,    253 ;    of 

frustum  of  regular  pyramid,  254  ; 

of  cone,  269  ;  of  frustum  of  regular 

cone,  269. 
Small  circle  of  sphere,  288. 
Solid  angle,  317  ;  measure  of,  317. 
Solids,    congruent,   242 ;    equivalent, 

245  ;  regular,  276. 
Sphere,  284  ;   center  of,  284  ;   radius 

of,  284  ;  diameter  of,  284  ;  tangent 

plane   to,    285 ;     tangent   line   to, 

285 ;    great  circle  of,   288 ;    small 

circle  of,   288 ;    distance   on,   288 ; 

inscribed,  294  ;  circumscribed,  294 ; 

area  of,  310  ;  volume  of,  318. 
Spherical  angle,  296. 
Spherical  degree,  315. 
Spherical  excess,  315. 
Spherical  polygon,  297. 
Spherical  triangle,  297  ;  area  of,  317. 
Symmetric  triangles,  305. 
Symmetric  trihedral  angles,  308. 

Tables,  i-xxvii. 
Tangent  line  to  sphere,  285. 
Tangent  plane,  285. 
Tetrahedron,  238  ;  similar,  275. 


INDEX 


xlix 


Triangle,  birectangular,  304 ;  spheri- 
cal, 297 ;  symmetric,  305 ;  tri- 
rectangular,  304. 

Trihedral  angle,  232  ;  congruent,  232  ; 
measure  of,  315. 

Trirectangular  triangle,  304. 

Truncated  prism,  243. 

Truncated  pyramid,  254. 

Vertex,  of  polyhedral  angle,  232 ; 
of  pyramid,  253  ;  of  spherical  poly- 
gon, 297. 


Volume,  of  rectangular  parallele- 
piped, 248  ;  of  any  parallelepiped, 
248  ;  of  triangular  prism,  251 ;  of 
any  prism,  251  ;  of  triangular 
pyramid,  260 ;  of  any  pyramid, 
261  ;  of  frustum  of  pyramid,  261, 
262;  of  any  cylinder,  266;  of 
circular  cylinder,  266  ;  of  any  cone, 
272  ;  of  frustum  of  cone,  272  ;  of 
sphere,  318. 

Zone,  of  sphere,  311  ;  area  of,  311. 


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